JP4821684B2 - Encoding device and encoding program - Google Patents

Description

  The present invention relates to an encoding device and an encoding program, and more particularly to an encoding device and an encoding program that generate a low density parity check code.

  When constructing a data communication system, high-speed communication, low power consumption, high communication quality (low bit error rate), and the like are required. An error correction technique for detecting and correcting an error in a received code is widely used in wireless, wired and recording systems as one technique that satisfies these requirements.

  In recent years, low density parity check (LDPC) codes and sum-product decoding methods have attracted attention as one of the error correction techniques. Decoding operation using this LDPC code is discussed in Non-Patent Document 1. This Non-Patent Document 1 shows that a decoding characteristic of 0.004 dB can be obtained up to the Shannon limit of a white Gaussian channel using an irregular LDPC code with a coding rate of 1/2. The irregular LDPC code indicates a code in which the row weight (number where 1 stands in a row) and the column weight (number where 1 stands in a column) of the parity check matrix are not constant. An LDPC code whose row weights and column weights are constant in each row and each column is called a regular LDPC code.

A technique for reducing the processing load of the LDPC encoding process, that is, the process of calculating the parity has been studied. For example, Non-Patent Document 2 discloses an encoding method using an approximate triangular matrix. Non-Patent Document 3 discloses an encoding method using triangulation. In these encoding methods, parity can be obtained by solving an equation including a lower triangular matrix and an upper triangular matrix.
SY Chung et al., “On the Design of Low-Density Parity-Check Codes within 0.0045dB of the Shannon Limit”, IEEE COMUNICATIONS LETTERS, VOL.5, No.2, Feb. 2001, pp.58-60 Thomas J. Richardson et al., “Efficient Encoding of Low-Density Parity-Check Codes”, IEEE TRANSACTIONS ON INFORMATION THEORY, VOL.47, No.2, Feb. 2001, pp.638-656 Yuichi Tsuji, “Computational Complexity of LDPC Coding Using Triangular Decomposition”, The 27th Symposium on Information Theory and Its Applications (SITA2004) Gero, Gifu, Japan, Dec 14-17, 2004

  However, in the encoding methods of Non-Patent Document 2 and Non-Patent Document 3, since the backward substitution or the like is used for solving the equations, the amount of calculation increases.

  Therefore, an object of the present invention is to provide an encoding apparatus and an encoding program capable of calculating parity with a smaller amount of calculation than conventional.

An encoding apparatus according to an aspect of the present invention performs an operation modulo 2 from an information vector X having the number of elements k according to a parity generation expression represented by the formula (A1), whereby the parity vector Y having the number of elements m. An encoding device for generating
Y≡U ₁ U ₂ ... U _m-1 U _m L _m L _m-1 ... L ₂ L ₁ AX (A1)
However, (X, Y) ^T represents transposition of the vector (X, Y), the parity check matrix H is an m-row n-column matrix satisfying H (X, Y) ^T ≡0, and the matrix A is , The parity check matrix H includes the first to k-th columns, and the matrix B includes the (k + 1) -th to n-th columns of the parity check matrix H, where n = k + m, and the matrix L _i (I = 1 to m) is a basic lower triangular matrix of the lower triangular matrix L obtained by LU decomposition of the matrix B. The i-th column matches the i-th column of the lower triangular matrix L, and the other columns are A matrix whose off-diagonal term is 0 and whose diagonal term is 1, and the matrix U _i (i = 1 to m) is a basic upper triangular matrix of the upper triangular matrix U obtained by LU decomposition of the matrix B, and The i-th row coincides with the i-th row of the upper triangular matrix U, and the other rows are matrices in which the off-diagonal term is 0 and the diagonal term is 1, and the encoding device is included in the parity generation formula. In order to calculate a parity vector Y, a matrix storage unit that stores data representing a plurality of matrices, an intermediate result storage unit that stores data representing intermediate results of a plurality of matrix multiplications included in the parity generation formula, A matrix product calculation unit that executes a plurality of matrix multiplications included in the parity generation formula, and the matrix product calculation unit receives the information vector X, data representing a matrix in the matrix storage unit, or an intermediate result in the intermediate result storage unit The data representing the calculation result is represented as an intermediate result until each matrix multiplication included in the parity generation formula is executed using the data to be represented and a plurality of matrix multiplications included in the parity generation formula is completed. When data is stored in the intermediate result storage unit and multiple matrix multiplications included in the parity generation formula are completed, after multiple matrix multiplications included in the parity generation formula are completed The obtained vector is output as a parity vector Y.

Preferably, the encoding device is an encoding device that generates a matrix included in the parity generation equation before generating the parity vector Y, and the intermediate result storage unit further converts the matrix included in the parity generation equation. Data representing an intermediate result of the process to be generated is stored, and the encoding apparatus further decomposes the parity check matrix H into a matrix A and a matrix B, and stores the data representing the matrix A in the matrix storage unit An LU decomposition unit for LU-decomposing the matrix B into a product of a lower triangular matrix L and an upper triangular matrix U and storing data representing the lower triangular matrix L and data representing the upper triangular matrix U in the intermediate result storage unit And reading data representing the upper triangular matrix U and the lower triangular matrix L from the intermediate result storage unit, and decomposing the upper triangular matrix U into a product of m basic upper triangular matrices U _i (i = 1 to m), M lower triangular matrix L Decomposing the product of the basic lower triangular matrix L _{i (i} = 1~m), decomposed matrix U _{i (i} = 1~m) and matrix L _{i (i} = 1~m) matrix storage unit data representing a And a basic matrix decomposition unit to be stored.

An encoding apparatus according to another aspect of the present invention generates a parity vector Y having m elements by performing a real number operation from an information vector X having k elements in accordance with a parity generation expression represented by Expression (A2). An encoding device comprising:
Y = −U ₁ ⁻¹ U ₂ ⁻¹ ... U _m-1 ⁻¹ U _m ⁻¹ L _m ⁻¹ L _m−1 ⁻¹ ... L ₂ ⁻¹ L ₁ ⁻¹ AX. A2)
However, (X, Y) ^T represents transposition of the vector (X, Y), the parity check matrix H is an m-by-n matrix satisfying H (X, Y) ^T = 0, and the matrix A is , The parity check matrix H includes the first to k-th columns, and the matrix B includes the (k + 1) -th to n-th columns of the parity check matrix H, where n = k + m, and the matrix L _i ^-1 (i = 1 to m) is the inverse matrix of the matrix L _i, the matrix L _i is a matrix B a fundamental lower triangular matrix of the lower triangular matrix L obtained by the LU decomposition, the i-th column is lower triangular matrix The other columns correspond to the i-th column of L, the off-diagonal term is 0, and the diagonal term is 1, and the matrix U _i ^-1 (i = ^{1 to} m) is the inverse of the matrix U _i . The matrix U _i is a basic upper triangular matrix of the upper triangular matrix U obtained by LU decomposition of the matrix B. The i-th row matches the i-th row of the upper triangular matrix U, and the other rows are Off-diagonal term is 0 , A matrix whose diagonal term is 1, and the encoding device includes: a matrix storage unit that stores data representing a plurality of matrices included in the parity generation formula; and a plurality of matrix multiplications included in the parity generation formula. An intermediate result storage unit that stores data representing the intermediate result, a matrix product calculation unit that executes matrix multiplication of a plurality of times included in the parity generation formula to calculate the parity vector Y, and a matrix product calculation unit are output. A matrix inversion unit that inverts the sign of the vector to output as a parity vector Y, and the matrix product calculation unit represents the information vector X, the data representing the matrix in the matrix storage unit, or the intermediate result in the intermediate result storage unit Data is used to represent the calculation result until each matrix multiplication included in the parity generation formula is executed and a plurality of matrix multiplications included in the parity generation formula are completed. Is stored in the intermediate result storage unit as data representing the intermediate result, and when the multiple matrix multiplications included in the parity generation formula are completed, the vector obtained after the multiple matrix multiplications included in the parity generation formula is inverted. To the output.

Preferably, the encoding device is an encoding device that generates a matrix included in the parity generation equation before generating the parity vector Y, and the intermediate result storage unit further converts the matrix included in the parity generation equation. Data representing an intermediate result of the process to be generated is stored, and the encoding apparatus further decomposes the parity check matrix H into a matrix A and a matrix B, and stores the data representing the matrix A in the matrix storage unit An LU decomposition unit for LU-decomposing the matrix B into a product of a lower triangular matrix L and an upper triangular matrix U and storing data representing the lower triangular matrix L and data representing the upper triangular matrix U in the intermediate result storage unit And reading data representing the upper triangular matrix U and the lower triangular matrix L from the intermediate result storage unit, and decomposing the upper triangular matrix U into a product of m basic upper triangular matrices U _i (i = 1 to m), M lower triangular matrix L Decomposing the product of the basic lower triangular matrix L _{i (i} = 1~m), decomposed matrix U _{i (i} = 1~m) and matrix L _{i (i} = 1~m) intermediate result storing data representing the a base matrix decomposition unit to be stored in the part, reads the data representative of the matrix from the intermediate result storage unit U _{i (i} = 1~m) and matrix L _{i (i} = 1~m), matrix U _{i (i} = 1 ˜m) and the matrix L _i (i = 1 to m) by inverting the signs of the elements of the non-diagonal terms, the inverse matrix U _i ⁻¹ (i = ^{1 to} m) and the inverse matrix L _i ⁻¹ ( i = ^{1 to} m) and inverse matrix calculation for storing data representing the inverse matrix U _i ⁻¹ (i = ^{1 to} m) and the inverse matrix L _i ⁻¹ (i = ^{1 to} m) in the matrix storage unit. A part.

The encoding apparatus according to another aspect of the present invention is based on an information vector X having k elements, and when u ≧ 1 and v ≧ 1, Expression (A3-1), and when u ≧ 1 and v = 0, Expression (A3) -2) Coding for generating a parity vector Y having m elements by performing an operation modulo 2 in accordance with the parity generation expression represented by Expression (A3-3) when u = 0 and v ≧ 1 A device,
Y≡E ^(v) U ′ ₁ U ′ ₂ ... U ′ _m−1 U ′ _m L ′ _m L ′ _m−1 ... L ′ ₂ L ′ ₁ A ^(u) X (A3 -1)
Y≡U ′ ₁ U ′ ₂ ... U ′ _m−1 U ′ _m L ′ _m L ′ _m−1 ... L ′ ₂ L ′ ₁ A ^(u) X (A3-2)
Y≡E ^(v) U ′ ₁ U ′ ₂ ... U ′ _m−1 U ′ _m L ′ _m L ′ _m−1 ... L ′ ₂ L ′ ₁ AX (A3-3)
However, (X, Y) ^T represents transposition of the vector (X, Y), the parity check matrix H is an m-row n-column matrix satisfying H (X, Y) ^T ≡0, and the matrix A is , The first column to the k-th column of the parity check matrix H, the matrix B includes the (k + 1) -th column to the n-th column of the parity check matrix H, n = k + m, and u ≧ 1 When v ≧ 1, the matrix B ′ is obtained by performing u times of row replacement and v times of column replacement with respect to the matrix B, and the matrix A ^(u) is the matrix A On the other hand, the same replacement as the u-time replacement of the matrix B is performed, and the matrix E ^(v) is the same as the v-time replacement of the matrix B with respect to the unit matrix E. If u ≧ 1 and v = 0, the matrix B ′ is replaced u times with respect to the matrix B when u ≧ 1 and v = 0. The matrix A ^(u) is obtained by performing the same replacement as the u-time replacement of the matrix B with respect to the matrix A when u = 0 and v ≧ 1. The matrix B ′ is obtained by performing v times of column replacement with respect to the matrix B, and the matrix E ^(v) is obtained by performing v times of column replacement for the matrix B with respect to the unit matrix E. The matrix L ′ _i (i = 1 to m) is a basic lower triangular matrix of the lower triangular matrix L ′ obtained by LU decomposition of the matrix B ′, and the i-th column is the lower one. The other columns correspond to the i-th column of the triangular matrix L ′, the off-diagonal term is 0, and the diagonal term is 1. The matrix U ′ _i (i = 1 to m) is the matrix B Is a basic upper triangular matrix of the upper triangular matrix U ′ obtained by LU decomposition of ′, the i-th row coincides with the i-th row of the upper triangular matrix U ′, and the off-diagonal term is 0 in the other rows. A matrix whose diagonal term is 1, the encoding device includes: a matrix storage unit that stores data representing a plurality of matrices included in the parity generation formula; and an intermediate between a plurality of matrix multiplications included in the parity generation formula An intermediate result storage unit that stores data representing the result, and a matrix product calculation unit that executes a plurality of matrix multiplications included in the parity generation formula in order to calculate the parity vector Y, Using the information vector X, the data representing the matrix in the matrix storage unit, or the data representing the intermediate result in the intermediate result storage unit, each matrix multiplication included in the parity generation formula is executed, and multiple times included in the parity generation formula Until the matrix multiplication is completed, the data representing the calculation result is stored in the intermediate result storage unit as data representing the intermediate result, and multiple times of matrix multiplication included in the parity generation formula are performed. When the calculation is completed, a vector obtained after the completion of a plurality of matrix multiplications included in the parity generation formula is output as a parity vector Y.

Preferably, the encoding device is an encoding device that generates a matrix included in the parity generation equation before generating the parity vector Y, and the intermediate result storage unit further converts the matrix included in the parity generation equation. Data representing an intermediate result of the process to be generated is stored, and the encoding apparatus further decomposes the parity check matrix H into a matrix A and a matrix B, and when u = 0 and v ≧ 1, data representing the matrix A And a parity check matrix decomposition unit that stores the matrix storage unit, and when u ≧ 1 and v ≧ 1, the matrix B is ordered, and u rows are exchanged and v columns are exchanged for the matrix B. A matrix B ′ is generated, and a matrix A ^{(u) in} which the same replacement as the u-time replacement of the matrix B is performed on the matrix A is generated and stored in the matrix storage unit. V times for matrix B A matrix E ^(v) having the same replacement as the column replacement is generated and stored in the matrix storage unit. When u ≧ 1 and v = 0, the matrix B is ordered, and u times for the matrix B Matrix B ′ is generated, and a matrix A ^{(u) in} which the same replacement as the u-time replacement of the matrix B is performed on the matrix A is generated and stored in the matrix storage unit. , U = 0 and v ≧ 1, the matrix B is ordered to generate a matrix B ′ in which the columns B are exchanged v times, and the unit matrix E is v times with respect to the matrix B. An LU decomposition into an ordering unit that generates a matrix E ^(v) that is exchanged in the same manner as the column exchange and stores the matrix E ^(v) in the matrix storage unit, and a matrix B ′ that is a product of the lower triangular matrix L ′ and the upper triangular matrix U ′ The data representing the lower triangular matrix L ′ and the upper triangular matrix U ′ are represented as LU decomposition unit for storing the data in the intermediate result storage unit, and data representing the upper triangular matrix U ′ and the lower triangular matrix L ′ are read from the intermediate result storage unit, and the upper triangular matrix U ′ is converted into m basic upper triangles. The matrix U ′ _i (i = 1 to m) is decomposed into products, the lower triangular matrix L ′ is decomposed into the products of m basic lower triangular matrices L ′ _i (i = 1 to m), and the decomposed matrix A basic matrix decomposing unit that stores data representing U ′ _i (i = 1 to m) and matrix′L _i (i = 1 to m) in the matrix storage unit.

The encoding apparatus according to an aspect of the present invention is based on an information vector X having k elements, and when u ≧ 1 and v ≧ 1, Expression (A4-1), and when u ≧ 1 and v = 0, Expression (A4- 2) When u = 0 and v ≧ 1, an encoding device that generates a parity vector Y having m elements by performing an operation modulo 2 in accordance with the parity generation expression represented by the expression (A4-3) Because
^{Y = -E (v) U '} 1 -1 U' 2 -1 ··· U 'm-1 -1 U' m -1 L 'm -1 L' m-1 -1 ··· L '2 ^-1 L ' ₁ ^-1 A ^(u) X (A4-1)
Y = -U ' ₁ ^-1 U' ₂ ^-1 ... U ' _m-1 ^-1 U' _m ^-1 L ' _m ^-1 L' _m-1 ^-1 ... L ' ₂ ^-1 L' ₁ ^-1 A ^(V) X (A4-2)
^{Y = -E (v) U '} 1 -1 U' 2 -1 ··· U 'm-1 -1 U' m -1 L 'm -1 L' m-1 -1 ··· L '2 ^-1 L ' ₁ ^-1 AX (A4-3)
However, the parity check matrix H is a matrix of m rows and n columns satisfying H (X, Y) ^T = 0, and the matrix A is composed of the first to kth columns of the parity check matrix H. B consists of the (k + 1) th column to the nth column of the parity check matrix H, and n = k + m. When u ≧ 1 and v ≧ 1, the matrix B ′ , U times of row replacement, and v times of column replacement, and the matrix A ^(u) performs the same replacement for the matrix A as u times of row replacement for the matrix B. The matrix E ^(v) is obtained by performing the same replacement as the unit matrix E with v column replacements for the matrix B. When u ≧ 1 and v = 0, , the matrix B ', to the matrix B, and which was subjected to replacement of u times the line, the matrix a ^(u), to the matrix a This is the same as the u-th row replacement for the matrix B. When u = 0 and v ≧ 1, the matrix B ′ replaces the matrix B with v columns. The matrix E ^(v) is obtained by performing the same permutation of the unit matrix E as the v-th permutation of the matrix B and the matrix L ′ _i ⁻¹ (i = 1). ~m) is 'is the inverse matrix of _i, the matrix L' matrix L _i is the matrix B a fundamental lower triangular matrix of 'the lower triangular matrix L obtained by the LU decomposition', the i-th column lower triangular matrix L 'And the other column is a matrix whose off-diagonal term is 0 and whose diagonal term is 1, and the matrix U' _i ^-1 (i = ^{1 to} m) is the matrix U ' _i of an inverse matrix, the matrix U _'i is a matrix B' to 'a basic upper triangular matrix, the i-th row upper triangular matrix U' upper triangular matrix U obtained by the LU decomposition coincides with the i-th row , And the other rows are matrices having off-diagonal terms of 0 and diagonal terms of 1, and the encoding device includes a matrix storage unit that stores data representing a plurality of matrices included in the parity generation formula, and a parity An intermediate result storage unit for storing data representing an intermediate result of a plurality of matrix multiplications included in the generation formula, and a matrix product for performing a plurality of matrix multiplications included in the parity generation formula to calculate a parity vector Y A calculation unit, and a code inversion unit that inverts the sign of the vector output from the matrix product calculation unit and outputs the inverted vector as a parity vector Y. The matrix product calculation unit includes information vector X and data representing a matrix in the matrix storage unit. Or, each matrix multiplication included in the parity generation formula is executed using the data representing the intermediate result in the intermediate result storage unit, and a plurality of matrix multiplications included in the parity generation formula are completed. Stores the data representing the calculation result in the intermediate result storage unit as data representing the intermediate result, and when the matrix multiplication of a plurality of times included in the parity generation formula is completed, the matrix multiplication of the plurality of times included in the parity generation formula is performed. The vector obtained after the completion is output to the sign inverting unit. The vector obtained after the completion of the multiple matrix multiplications included is output to the sign inverting unit.

Preferably, the encoding device is an encoding device that generates a matrix included in the parity generation equation before generating the parity vector Y, and the intermediate result storage unit further converts the matrix included in the parity generation equation. Data representing an intermediate result of the process to be generated is stored, and the encoding apparatus further decomposes the parity check matrix H into a matrix A and a matrix B, and when u = 0 and v ≧ 1, data representing the matrix A And a parity check matrix decomposition unit that stores the matrix storage unit, and when u ≧ 1 and v ≧ 1, the matrix B is ordered, and u rows are exchanged and v columns are exchanged for the matrix B. A matrix B ′ is generated, and a matrix A ^{(u) in} which the same replacement as the u-time replacement of the matrix B is performed on the matrix A is generated and stored in the matrix storage unit. V times for matrix B A matrix E ^(v) having the same replacement as the column replacement is generated and stored in the matrix storage unit. When u ≧ 1 and v = 0, the matrix B is ordered, and u times for the matrix B Matrix B ′ is generated, and a matrix A ^{(u) in} which the same replacement as the u-time replacement of the matrix B is performed on the matrix A is generated and stored in the matrix storage unit. , U = 0 and v ≧ 1, the matrix B is ordered to generate a matrix B ′ in which the columns B are exchanged v times, and the unit matrix E is v times with respect to the matrix B. An LU decomposition into an ordering unit that generates a matrix E ^(v) that is exchanged in the same manner as the column exchange and stores the matrix E ^(v) in the matrix storage unit, and a matrix B ′ that is a product of the lower triangular matrix L ′ and the upper triangular matrix U ′ The data representing the lower triangular matrix L ′ and the upper triangular matrix U ′ are represented as An LU decomposition unit for storing the data in the intermediate result storage unit, and data representing the upper triangular matrix U ′ and the lower triangular matrix L ′ are read from the intermediate result storage unit, and the upper triangular matrix U ′ is converted into m basic upper triangles. The matrix U ′ _i (i = 1 to m) is decomposed into products, the lower triangular matrix L ′ is decomposed into the products of m basic lower triangular matrices L ′ _i (i = 1 to m), and the decomposed matrix A basic matrix decomposition unit that stores data representing U ′ _i (i = 1 to m) and matrix ′ L _i (i = 1 to m) in the intermediate result storage unit, and a matrix U ′ _i (i = 1 to m) and the data representing the matrix L ′ _i (i = 1 to m) are read, and the matrix U ′ _i (i = 1 to m) and the matrix L ′ _i (i = 1 to m) are unpaired. by reversing the sign of the elements of Sumiko calculates the inverse matrix ^{_{U 'i -1 (i = 1~m}} ) and the inverse matrix ^{_{L' i -1 (i = 1~m}} ) And an inverse matrix ^{_{U 'i -1 (i = 1~m}} ) and the inverse matrix ^{_{L' i -1 (i = 1~m}} ) inverse matrix calculating unit for storing data in the matrix storage unit representing the.

An encoding apparatus according to another aspect of the present invention provides a multiplication result of two or more matrices obtained by multiplying at least one set of adjacent matrices included in Equation (A5) from information vector X having k elements. An encoding device that generates a parity vector Y having m elements by performing an operation modulo 2 in accordance with a parity generation expression replaced by a matrix C,
Y≡U ₁ U ₂ ... U _m-1 U _m L _m L _m-1 ... L ₂ L ₁ AX (A5)
However, (X, Y) ^T represents transposition of the vector (X, Y), the parity check matrix H is an m-row n-column matrix satisfying H (X, Y) ^T ≡0, and the matrix A is , The parity check matrix H includes the first to k-th columns, and the matrix B includes the (k + 1) -th to n-th columns of the parity check matrix H, where n = k + m, and the matrix L _i (I = 1 to m) is a basic lower triangular matrix of the lower triangular matrix L obtained by LU decomposition of the matrix B. The i-th column matches the i-th column of the lower triangular matrix L, and the other columns are A matrix whose off-diagonal term is 0 and whose diagonal term is 1, and the matrix U _i (i = 1 to m) is a basic upper triangular matrix of the upper triangular matrix U obtained by LU decomposition of the matrix B, and The i-th row coincides with the i-th row of the upper triangular matrix U, and the other rows are matrices in which the off-diagonal term is 0 and the diagonal term is 1, and the encoding device is included in the parity generation formula. In order to calculate a parity vector Y, a matrix storage unit that stores data representing a plurality of matrices, an intermediate result storage unit that stores data representing intermediate results of a plurality of matrix multiplications included in the parity generation formula, A matrix product calculation unit that executes a plurality of matrix multiplications included in the parity generation formula, and the matrix product calculation unit receives the information vector X, data representing a matrix in the matrix storage unit, or an intermediate result in the intermediate result storage unit The data representing the calculation result is represented as an intermediate result until each matrix multiplication included in the parity generation formula is executed using the data to be represented and a plurality of matrix multiplications included in the parity generation formula is completed. When data is stored in the intermediate result storage unit and multiple matrix multiplications included in the parity generation formula are completed, after multiple matrix multiplications included in the parity generation formula are completed The obtained vector is output as a parity vector Y.

  Preferably, the sum of the number of non-zero elements of the non-diagonal terms of all the matrices included in the parity generation formula is less than or equal to the sum of the number of non-zero elements of the non-diagonal terms of all the matrices included in Formula (A5). is there.

Preferably, the encoding device is an encoding device that generates a matrix included in the parity generation equation before generating the parity vector Y, and the intermediate result storage unit further converts the matrix included in the parity generation equation. Data representing an intermediate result of the process to be generated is stored, and the matrix product calculation unit further performs multiplication of two or more adjacent matrices included in Expression (A5) to represent matrix C as a multiplication result. The encoding apparatus further decomposes the parity check matrix H into a matrix A and a matrix B and stores the data representing the matrix A in the matrix storage unit, and the matrix B as a lower triangle. LU decomposition into a product of the matrix L and the upper triangular matrix U and LU decomposition unit for storing data representing the lower triangular matrix L and data representing the upper triangular matrix U in the intermediate result storage unit; Matrix U and Reads the data representing the lower triangular matrix L, and the upper triangular matrix U was decomposed into a product of m elementary upper triangular matrix U _{i (i} = 1~m), the lower triangular matrix L m-number of basic lower triangular matrix L _i (i = 1 to m) is decomposed into products, and data representing the decomposed matrix U _i (i = 1 to m) and the matrix L _i (i = 1 to m) is stored in the intermediate result storage unit. The basic matrix decomposition unit, the number of non-zero elements of the matrix C as a multiplication result, and the number of non-zero elements of the non-diagonal terms of the matrix excluding two or more adjacent matrices among the matrices included in the formula (A5) Of the matrices included in the equation (A5) stored in the intermediate result storage unit when the sum is less than or equal to the sum of the number of nonzero elements in the non-diagonal terms of all the matrices included in the equation (A5) Matrix multiplication factor reduction storing data representing a matrix excluding two or more adjacent matrices and data representing a matrix C in a matrix storage unit. And a part.

An encoding apparatus according to another aspect of the present invention provides a multiplication result of two or more matrices obtained by multiplying at least one set of adjacent matrices included in Equation (A6) from information vector X having k elements. An encoding device that generates a parity vector Y having m elements by performing a real number operation according to a parity generation expression replaced by a matrix C,
Y = −U ₁ ⁻¹ U ₂ ⁻¹ ... U _m-1 ⁻¹ U _m ⁻¹ L _m ⁻¹ L _m−1 ⁻¹ ... L ₂ ⁻¹ L ₁ ⁻¹ AX. A6)
However, (X, Y) ^T represents transposition of the vector (X, Y), the parity check matrix H is an m-by-n matrix satisfying H (X, Y) ^T = 0, and the matrix A is , The parity check matrix H includes the first to k-th columns, and the matrix B includes the (k + 1) -th to n-th columns of the parity check matrix H, where n = k + m, and the matrix L _i ^-1 (i = 1 to m) is the inverse matrix of the matrix L _i, the matrix L _i is a matrix B a fundamental lower triangular matrix of the lower triangular matrix L obtained by the LU decomposition, the i-th column is lower triangular matrix The other columns correspond to the i-th column of L, the off-diagonal term is 0, and the diagonal term is 1, and the matrix U _i ^-1 (i = ^{1 to} m) is the inverse of the matrix U _i . The matrix U _i is a basic upper triangular matrix of the upper triangular matrix U obtained by LU decomposition of the matrix B. The i-th row matches the i-th row of the upper triangular matrix U, and the other rows are Off-diagonal term is 0 , A matrix whose diagonal term is 1, and the encoding device includes: a matrix storage unit that stores data representing a plurality of matrices included in the parity generation formula; and a plurality of matrix multiplications included in the parity generation formula. An intermediate result storage unit that stores data representing the intermediate result, a matrix product calculation unit that executes matrix multiplication of a plurality of times included in the parity generation formula to calculate the parity vector Y, and a matrix product calculation unit are output. A matrix inversion unit that inverts the sign of the vector to output as a parity vector Y, and the matrix product calculation unit represents the information vector X, the data representing the matrix in the matrix storage unit, or the intermediate result in the intermediate result storage unit Data is used to represent the calculation result until each matrix multiplication included in the parity generation formula is executed and a plurality of matrix multiplications included in the parity generation formula are completed. Is stored in the intermediate result storage unit as data representing the intermediate result, and when the multiple matrix multiplications included in the parity generation formula are completed, the vector obtained after the multiple matrix multiplications included in the parity generation formula is inverted. To the output.

  Preferably, the sum of the non-zero element numbers of the non-diagonal terms of all the matrices included in the parity generation equation is equal to or less than the sum of the non-zero element numbers of the non-diagonal terms of all the matrices included in the equation (A6). is there.

Preferably, the encoding device is an encoding device that generates a matrix included in the parity generation equation before generating the parity vector Y, and the intermediate result storage unit further converts the matrix included in the parity generation equation. Data representing an intermediate result of the process to be generated is stored, and the matrix product calculation unit further performs multiplication of two or more adjacent matrices included in Expression (A6) to represent a matrix C that is a multiplication result. The encoding apparatus further decomposes the parity check matrix H into a matrix A and a matrix B and stores the data representing the matrix A in the matrix storage unit, and the matrix B as a lower triangle. LU decomposition into a product of the matrix L and the upper triangular matrix U and LU decomposition unit for storing data representing the lower triangular matrix L and data representing the upper triangular matrix U in the intermediate result storage unit; Matrix U and Reads the data representing the lower triangular matrix L, and the upper triangular matrix U was decomposed into a product of m elementary upper triangular matrix U _{i (i} = 1~m), the lower triangular matrix L m-number of basic lower triangular matrix L _i (i = 1 to m) is decomposed into products, and data representing the decomposed matrix U _i (i = 1 to m) and the matrix L _i (i = 1 to m) is stored in the intermediate result storage unit. Data representing the matrix U _i (i = 1 to m) and the matrix L _i (i = 1 to m) are read from the basic matrix decomposition unit and the intermediate result storage unit, and the matrix U _i (i = 1 to m) and by reversing the sign of the non-diagonal terms elements of the matrix L _{i (i} = 1~m), inverse matrix U _i ^-1 _(i = 1~m) and the inverse matrix L _i ^-1 _(i = 1~ m) calculating the inverse matrix U _i ^-1 _(i = 1~m) and the inverse matrix L _i ^-1 _(i = 1~m) inverse matrix calculating unit for storing data in the intermediate result storage unit representing the , The sum of the number of non-zero elements of the matrix C, which is the multiplication result, and the number of non-zero elements of the non-diagonal terms of the matrix excluding two or more adjacent matrices among the matrices included in the expression (A6), Two or more adjacent matrices among the matrices included in the expression (A6) stored in the intermediate result storage unit when the number of non-zero elements of the non-diagonal terms of all the matrices included in A6) is less than or equal to A matrix multiplication number reduction unit that stores data representing the matrix excluding the matrix and data representing the matrix C in the matrix storage unit.

The encoding device according to another aspect of the present invention is based on an information vector X having k elements, and when u ≧ 1 and v ≧ 1, Expression (A7-1), and when u ≧ 1 and v = 0, Expression (A7) -2) When u = 0 and v ≧ 1, at least one set of two or more adjacent matrices included in the formula (A7-3) is replaced with a matrix C which is a multiplication result of two or more matrices. An encoding device that generates a parity vector Y having m elements by performing an operation modulo 2 according to the parity generation equation,
Y≡E ^(v) U ′ ₁ U ′ ₂ ... U ′ _m−1 U ′ _m L ′ _m L ′ _m−1 ... L ′ ₂ L ′ ₁ A ^(u) X (A7 -1)
Y≡U ′ ₁ U ′ ₂ ... U ′ _m-1 U ′ _m L ′ _m L ′ _m−1 ... L ′ ₂ L ′ ₁ A ^(u) X (A7-2)
Y≡E ^(v) U ′ ₁ U ′ ₂ ... U ′ _m−1 U ′ _m L ′ _m L ′ _m−1 ... L ′ ₂ L ′ ₁ AX (A7-3)
However, (X, Y) ^T represents transposition of the vector (X, Y), the parity check matrix H is an m-row n-column matrix satisfying H (X, Y) ^T ≡0, and the matrix A is , The first column to the k-th column of the parity check matrix H, the matrix B includes the (k + 1) -th column to the n-th column of the parity check matrix H, n = k + m, and u ≧ 1 When v ≧ 1, the matrix B ′ is obtained by performing u times of row replacement and v times of column replacement with respect to the matrix B, and the matrix A ^(u) is the matrix A On the other hand, the same replacement as the u-time replacement of the matrix B is performed, and the matrix E ^(v) is the same as the v-time replacement of the matrix B with respect to the unit matrix E. If u ≧ 1 and v = 0, the matrix B ′ is replaced u times with respect to the matrix B when u ≧ 1 and v = 0. The matrix A ^(u) is obtained by performing the same replacement as the u-time replacement of the matrix B with respect to the matrix A when u = 0 and v ≧ 1. The matrix B ′ is obtained by performing v times of column replacement with respect to the matrix B, and the matrix E ^(v) is obtained by performing v times of column replacement for the matrix B with respect to the unit matrix E. The matrix L ′ _i (i = 1 to m) is a basic lower triangular matrix of the lower triangular matrix L ′ obtained by LU decomposition of the matrix B ′, and the i-th column is the lower one. The other columns correspond to the i-th column of the triangular matrix L ′, the off-diagonal term is 0, and the diagonal term is 1. The matrix U ′ _i (i = 1 to m) is the matrix B Is a basic upper triangular matrix of the upper triangular matrix U ′ obtained by LU decomposition of ′, the i-th row coincides with the i-th row of the upper triangular matrix U ′, and the off-diagonal term is 0 in the other rows. A matrix whose diagonal term is 1, the encoding device includes: a matrix storage unit that stores data representing a plurality of matrices included in the parity generation formula; and an intermediate between a plurality of matrix multiplications included in the parity generation formula An intermediate result storage unit that stores data representing the result, and a matrix product calculation unit that executes a plurality of matrix multiplications included in the parity generation formula in order to calculate the parity vector Y, Using the information vector X, the data representing the matrix in the matrix storage unit, or the data representing the intermediate result in the intermediate result storage unit, each matrix multiplication included in the parity generation formula is executed, and multiple times included in the parity generation formula Until the matrix multiplication is completed, the data representing the calculation result is stored in the intermediate result storage unit as data representing the intermediate result, and multiple times of matrix multiplication included in the parity generation formula are performed. When the calculation is completed, a vector obtained after the completion of a plurality of matrix multiplications included in the parity generation formula is output as a parity vector Y.

  Preferably, the sum of the number of non-zero elements of the non-diagonal terms of all the matrices included in the parity generation formula is the sum of all the numbers included in the formula (A7-1), (A7-2), or (A7-3). Is less than or equal to the sum of the number of nonzero elements in the off-diagonal terms of the matrix.

Preferably, the encoding device is an encoding device that generates a matrix included in the parity generation equation before generating the parity vector Y, and the intermediate result storage unit further converts the matrix included in the parity generation equation. Data representing an intermediate result of the process to be generated is stored, and the matrix product calculation unit further includes two or more adjacent matrices included in the formula (A7-1), (A7-2), or (A7-3) And the data representing the matrix C as the multiplication result is output, and the encoding apparatus further decomposes the parity check matrix H into the matrix A and the matrix B, and the data representing the matrix A and the matrix B Is stored in the intermediate result storage unit, and when u ≧ 1 and v ≧ 1, the data representing the matrix A and the matrix B is read from the intermediate result storage unit, the matrix B is ordered, and the matrix B Against u Permutations and v times to generate a matrix B 'was subjected to replacement column of the row, to generate a matrix A ^(u), which was carried out in the same manner as the replacement and swapping of u times the line for the matrix B for matrices A Are stored in the intermediate result storage unit, a matrix E ^{(v) in} which the same replacement as the column replacement for the matrix B is performed on the unit matrix E is generated and stored in the intermediate result storage unit, and u ≧ When 1 and v = 0, data representing the matrix A and the matrix B is read from the intermediate result storage unit, the matrix B is ordered, and the matrix B ′ in which u rows are replaced with respect to the matrix B is obtained. When the matrix A ^(u) is generated and the matrix A ^(u) is exchanged in the same manner as the matrix B with respect to the matrix B, and stored in the intermediate result storage unit, and u = 0 and v ≧ 1 Read data representing matrix B from the intermediate result storage Then, the matrix B is ordered to generate a matrix B ′ in which the columns B are exchanged v times for the matrix B, and the unit matrix E is exchanged similarly to the v times column exchange for the matrix B. An ordering unit that generates the matrix E ^(v) and stores the result in the intermediate result storage unit, and LU-decomposes the matrix B ′ into the product of the lower triangular matrix L ′ and the upper triangular matrix U ′, and lower matrix L Data representing ′ and data representing the upper triangular matrix U ′ are stored in the intermediate result storage unit, and data representing the upper triangular matrix U ′ and the lower triangular matrix L ′ are read from the intermediate result storage unit. The triangular matrix U ′ is decomposed into a product of m basic upper triangular matrices U ′ _i (i = 1 to m), and the lower triangular matrix L ′ is converted to m basic lower triangular matrices L ′ _i (i = 1 to m). decomposing the product of), decomposed matrix U _{'i (i} = 1~m) and matrix L _{i (i} = 1~m) and the base matrix decomposition unit for storing data in the intermediate result storage unit representing a non-zero number of elements and expressions of the matrix C is a multiplication result (A7-1), (A7-2) Or the sum of the number of non-zero elements in the non-diagonal terms of the matrix excluding two or more adjacent matrices among the matrices included in (A7-3) is expressed by equations (A7-1) and (A7-2). , Or (A7-3), the formulas (A7-1) and (A7−) stored in the intermediate result storage unit are equal to or less than the sum of the number of nonzero elements of the off-diagonal terms of all the matrices. 2) or a matrix multiplication number reduction unit that stores data representing a matrix excluding two or more adjacent matrices in the matrix included in (A7-3) and data representing a matrix C in a matrix storage unit; Is provided.

The encoding apparatus according to another aspect of the present invention is based on an information vector X having k elements, and is expressed by equation (A8-1) when u ≧ 1 and v ≧ 1, and equation (A8) when u ≧ 1 and v = 0. -2) When u = 0 and v ≧ 1, at least one set of two or more adjacent matrices included in equation (A8-3) is replaced with a matrix C that is the result of multiplication of two or more matrices. An encoding device that generates a parity vector Y having m elements by performing a real number operation according to the parity generation equation,
^{Y = -E (v) U '} 1 -1 U' 2 -1 ··· U 'm-1 -1 U' m -1 L 'm -1 L' m-1 -1 ··· L '2 ^-1 L ' ₁ ^-1 A ^(u) X (A8-1)
Y = -U ' ₁ ^-1 U' ₂ ^-1 ... U ' _m-1 ^-1 U' _m ^-1 L ' _m ^-1 L' _m-1 ^-1 ... L ' ₂ ^-1 L' ₁ ^-1 A ^(V) X (A8-2)
^{Y = -E (v) U '} 1 -1 U' 2 -1 ··· U 'm-1 -1 U' m -1 L 'm -1 L' m-1 -1 ··· L '2 ^-1 L ' ₁ ^-1 AX (A8-3)
However, (X, Y) ^T represents transposition of the vector (X, Y), the parity check matrix H is an m-by-n matrix satisfying H (X, Y) ^T = 0, and the matrix A is , The first column to the k-th column of the parity check matrix H, the matrix B includes the (k + 1) -th column to the n-th column of the parity check matrix H, n = k + m, and u ≧ 1 When v ≧ 1, the matrix B ′ is obtained by performing u times of row replacement and v times of column replacement with respect to the matrix B, and the matrix A ^(u) is the matrix A On the other hand, the same replacement as the u-time replacement of the matrix B is performed, and the matrix E ^(v) is the same as the v-time replacement of the matrix B with respect to the unit matrix E. If u ≧ 1 and v = 0, the matrix B ′ is replaced u times with respect to the matrix B when u ≧ 1 and v = 0. The matrix A ^(u) is obtained by performing the same replacement as the u-time replacement of the matrix B with respect to the matrix A when u = 0 and v ≧ 1. The matrix B ′ is obtained by performing v times of column replacement with respect to the matrix B, and the matrix E ^(v) is obtained by performing v times of column replacement for the matrix B with respect to the unit matrix E. are those conducted similar replacement, the matrix ^{_{L 'i -1 (i = 1~m}} ) is a matrix L' is the inverse matrix of _i, the matrix L _'i is a matrix B' lower triangular matrix obtained by the LU decomposition of A basic lower triangular matrix of L ′, in which the i-th column coincides with the i-th column of the lower triangular matrix L ′, and the other columns are matrices whose off-diagonal terms are 0 and whose diagonal terms are 1. There, the matrix ^{_{U 'i -1 (i = 1~m}} ) is a matrix U' is the inverse matrix of _i, the matrix U _'i is a matrix B' of the basic upper triangular matrix of the upper triangular matrix U 'was LU decomposition The i-th row matches the i-th row of the upper triangular matrix U ′, and the other rows are matrices whose off-diagonal terms are 0 and diagonal terms are 1. A matrix storage unit for storing data representing a plurality of matrices included in the generation formula, an intermediate result storage unit for storing data representing intermediate results of a plurality of matrix multiplications included in the parity generation formula, and a parity vector Y In order to calculate, a matrix product calculation unit that performs a plurality of matrix multiplications included in the parity generation equation, a code inversion unit that inverts the sign of the vector output from the matrix product calculation unit and outputs the result as a parity vector Y; The matrix product calculation unit executes each matrix multiplication included in the parity generation formula using the information vector X, the data representing the matrix in the matrix storage unit, or the data representing the intermediate result in the intermediate result storage unit. ,parity Until the multiple matrix multiplications included in the formula are completed, the data representing the calculation result is stored in the intermediate result storage unit as data representing the intermediate results, and the multiple matrix multiplications included in the parity generation formula are performed. Is completed, a vector obtained after the completion of the multiple matrix multiplications included in the parity generation equation is output to the sign inversion unit.

  Preferably, the sum of the number of non-zero elements of the non-diagonal terms of all the matrices included in the parity generation equation is calculated by adding all the numbers included in the equation (A8-1), (A8-2), or (A8-3). Is less than or equal to the sum of the number of nonzero elements in the off-diagonal terms of the matrix.

Preferably, the encoding device is an encoding device that generates a matrix included in the parity generation equation before generating the parity vector Y, and the intermediate result storage unit further converts the matrix included in the parity generation equation. Data representing an intermediate result of the process to be generated is stored, and the matrix product calculation unit further includes two or more adjacent matrices included in the formula (A8-1), (A8-2), or (A8-3) And the data representing the matrix C as the multiplication result is output, and the encoding apparatus further decomposes the parity check matrix H into the matrix A and the matrix B, and the data representing the matrix A and the matrix B Is stored in the intermediate result storage unit, and when u ≧ 1 and v ≧ 1, the data representing the matrix A and the matrix B is read from the intermediate result storage unit, the matrix B is ordered, and the matrix B Against u Permutations and v times to generate a matrix B 'was subjected to replacement column of the row, to generate a matrix A ^(u), which was carried out in the same manner as the replacement and swapping of u times the line for the matrix B for matrices A Are stored in the intermediate result storage unit, a matrix E ^{(v) in} which the same replacement as the column replacement for the matrix B is performed on the unit matrix E is generated and stored in the intermediate result storage unit, and u ≧ When 1 and v = 0, data representing the matrix A and the matrix B is read from the intermediate result storage unit, the matrix B is ordered, and the matrix B ′ in which u rows are replaced with respect to the matrix B is obtained. When the matrix A ^(u) is generated and the matrix A ^(u) is exchanged in the same manner as the matrix B with respect to the matrix B, and stored in the intermediate result storage unit, and u = 0 and v ≧ 1 Read data representing matrix B from the intermediate result storage Then, the matrix B is ordered to generate a matrix B ′ in which the columns B are exchanged v times for the matrix B, and the unit matrix E is exchanged similarly to the v times column exchange for the matrix B. An ordering unit that generates the matrix E ^(v) and stores the result in the intermediate result storage unit, and LU-decomposes the matrix B ′ into the product of the lower triangular matrix L ′ and the upper triangular matrix U ′, and lower matrix L Data representing ′ and data representing the upper triangular matrix U ′ are stored in the intermediate result storage unit, and data representing the upper triangular matrix U ′ and the lower triangular matrix L ′ are read from the intermediate result storage unit. The triangular matrix U ′ is decomposed into a product of m basic upper triangular matrices U ′ _i (i = 1 to m), and the lower triangular matrix L ′ is converted to m basic lower triangular matrices L ′ _i (i = 1 to m). decomposing the product of), decomposed matrix U _{'i (i} = 1~m) and matrix L _{i (i} = 1~m) and the base matrix decomposition unit for storing data in the intermediate result storage unit representing the matrix from the intermediate result storage unit U _{'i (i} = 1~m) and matrix L' _{i (i} = 1-m) is read out and inverted by inverting the signs of the elements of the off-diagonal terms of the matrix U ′ _i (i = 1-m) and the matrix L ′ _i (i = 1-m). The matrix U ′ _i ⁻¹ (i = ^{1 to} m) and the inverse matrix L ′ _i ⁻¹ (i = ^{1 to} m) are calculated, and the inverse matrix U ′ _i ⁻¹ (i = ^{1 to} m) and the inverse matrix L An inverse matrix calculation unit that stores data representing ′ _i ⁻¹ (i = ^{1 to} m) in the intermediate result storage unit, the number of non-zero elements of the matrix C that is the multiplication result, and equations (A8-1) and (A8− 2) or the sum of the number of non-zero elements in the non-diagonal terms of the matrix excluding two or more adjacent matrices among the matrices included in (A8-3) is expressed by equations (A8-1), (A8- 2), Or, when it is equal to or less than the sum of the number of non-zero elements of non-diagonal terms of all the matrices included in (A8-3), the expressions (A8-1) and (A8-2) stored in the intermediate result storage ) Or (A8-3), a matrix multiplication number reduction unit for storing data representing a matrix excluding two or more adjacent matrices and data representing a matrix C in a matrix storage unit Prepare.

  Preferably, the matrix product calculation unit sequentially executes the matrix multiplication on the right side of the parity generation formula among a plurality of matrix multiplications included in the parity generation formula.

  Preferably, the matrix product calculation unit executes two or more matrix multiplications in parallel among a plurality of matrix multiplications included in the parity generation formula.

An encoding program according to another aspect of the present invention performs an operation modulo 2 from an information vector X having the number of elements k according to a parity generation expression represented by the formula (A9), thereby generating a parity vector having the number of elements m. An encoding program for generating Y,
Y≡U ₁ U ₂ ... U _m-1 U _m L _m L _m-1 ... L ₂ L ₁ AX (A9)
However, (X, Y) ^T represents transposition of the vector (X, Y), the parity check matrix H is an m-row n-column matrix satisfying H (X, Y) ^T ≡0, and the matrix A is , The parity check matrix H includes the first to k-th columns, and the matrix B includes the (k + 1) -th to n-th columns of the parity check matrix H, where n = k + m, and the matrix L _i (I = 1 to m) is a basic lower triangular matrix of the lower triangular matrix L obtained by LU decomposition of the matrix B. The i-th column matches the i-th column of the lower triangular matrix L, and the other columns are A matrix whose off-diagonal term is 0 and whose diagonal term is 1, and the matrix U _i (i = 1 to m) is a basic upper triangular matrix of the upper triangular matrix U obtained by LU decomposition of the matrix B, and The i-th row coincides with the i-th row of the upper triangular matrix U, and the other rows are matrices in which the off-diagonal term is 0 and the diagonal term is 1, and the encoding program A matrix storage unit that stores data representing a plurality of matrices included in the parity generation equation, an intermediate result storage unit that stores data representing intermediate results of a plurality of matrix multiplications included in the parity generation equation, and a parity vector Y To calculate a matrix product calculation unit that performs a plurality of matrix multiplications included in the parity generation formula, and the matrix product calculation unit is an information vector X, data representing a matrix in the matrix storage unit, or an intermediate result The data representing the intermediate result in the storage unit is used to execute each matrix multiplication included in the parity generation formula, and the calculation result is expressed until a plurality of matrix multiplications included in the parity generation formula are completed. When data is stored in the intermediate result storage unit as data representing the intermediate result, and multiple matrix multiplications included in the parity generation formula are completed, the data is included in the parity generation formula. A vector obtained after completion of multiple matrix multiplications is output as a parity vector Y.

Preferably, the encoding program is an encoding program for generating a matrix included in the parity generation equation before generating the parity vector Y, and the intermediate result storage unit further converts the matrix included in the parity generation equation. Data representing an intermediate result of the process to be generated is stored, and the encoding program further decomposes the parity check matrix H into matrix A and matrix B, and stores the data representing matrix A in the matrix storage unit. LU that causes a parity check matrix decomposition unit to LU-decompose matrix B into a product of lower triangular matrix L and upper triangular matrix U, and store data representing lower triangular matrix L and data representing upper triangular matrix U in the intermediate result storage unit Data representing the upper triangular matrix U and the lower triangular matrix L are read from the decomposition unit and the intermediate result storage unit, and the upper triangular matrix U is converted into m basic upper triangular matrices U _i (i = 1 to m). The lower triangular matrix L is decomposed into products of m basic lower triangular matrices L _i (i = 1 to m), and the decomposed matrix U _i (i = 1 to m) and the matrix L _i are decomposed. It is made to function as a basic matrix decomposition unit for storing data representing (i = 1 to m) in the matrix storage unit.

  According to the present invention, parity can be calculated with a smaller amount of calculation than in the prior art.

(overall structure)
Hereinafter, embodiments according to the present invention will be described with reference to the drawings.

FIG. 1 is a diagram illustrating an example of a configuration of a communication system according to an embodiment of this invention.
In the communication system of FIG. 1, the transmission device 1 includes an encoder 2 and a modulator 3, and the reception device 4 includes a demodulator 6 and an error correction decoder 7.

  The encoder 2 adds a parity vector Y consisting of m-bit parity bits to an information vector X consisting of k-bit information bits, and an n (= (k + m))-bit LDPC code (low density parity check) A code Z is generated. In the parity check matrix H (m rows and n columns) used for generating the parity vector Y, the number of rows m corresponds to the number of redundant bits, and the number of columns n corresponds to the number of code bits.

  The modulator 3 performs modulation such as amplitude modulation, phase modulation, code modulation, frequency modulation, or direct frequency division multiplexing modulation on the LDPC code Z according to the configuration of the communication path 7.

  The demodulator 6 performs a demodulation process on the modulated signal received via the communication path 7 according to the transmission form on the communication path 7, thereby performing n-bit multilevel quantization (L is expressed as a multilevel number). Then, the digital code with L ≧ 2) is output. For example, in the case of amplitude modulation, phase modulation, code modulation, frequency modulation, and orthogonal frequency division multiplexing modulation, the demodulator performs processing such as amplitude demodulation, phase demodulation, code demodulation, and frequency demodulation.

  The error correction decoder 5 applies the parity check matrix H to the n-bit code output from the demodulator to restore the information vector X composed of k-bit information bits.

[First Embodiment]
The encoder of the first embodiment performs an operation modulo 2.

(Encoding algorithm)
The encoder receives an information vector X consisting of k information bits x1, x2,..., Xk and performs an operation modulo 2 so as to satisfy m A parity vector Y composed of bits y1, y2,.
H (X, Y) ^T ≡0 (B1)
Here, H is a parity check matrix (hereinafter also referred to as a check matrix), which is a matrix of m rows and n columns (= k + m).
(X, Y) ^T represents transposition of a vector composed of an information vector and a parity vector.

Expression (B1) is modified as follows in the case of an operation modulo 2.
BY≡AX (B2)
Here, the matrix A is a matrix of m rows and k columns including the first to kth columns of the check matrix H. The matrix B is a matrix of m rows and m columns including the (k + 1) -th column to the n-th column of the parity check matrix H.

When the matrix B is subjected to LU decomposition, the equation (B2) is transformed into the following equation.
LUY≡AX (B3)
Here, L is a lower triangular matrix and U is an upper triangular matrix.
From the equation (B3), the column vector Y is expressed by the following equation.
Y≡U ^-1 L ^-1 AX (B4)

The upper triangular matrix U is decomposed into products of basic upper triangular matrices U _i (i = 1 to m) as in the following equation.
U≡U _m U _m-1 ... U ₂ U ₁ ... (B5)
From Equation (B5), the inverse matrix U ⁻¹ is represented by the following equation.
U ^-1 ≡ U ₁ ^-1 U ₂ ^-1 ... U _m-1 ^-1 U _m ^-1 (B6)

The lower triangular matrix L is decomposed into products of basic lower triangular matrices L _i (i = 1 to m) as shown in the following expression.
L≡L ₁ L ₂ ... L _m-1 L _m (B7)
From the equation (B7), the inverse matrix L ⁻¹ is expressed by the following equation.
L ^-1 ≡L _m ^-1 L _m-1 ^-1 ... L ₂ ^-1 L ₁ ^-1 ... (B8)

Expression (B4) is transformed into the following expression by Expression (B6) and Expression (B8).
Y≡ (U ₁ ^-1 U ₂ ^-1 ... U _m-1 ^-1 U _m ^-1 ) (L _m ^-1 L _m-1 ^-1 ... L ₂ ^-1 L ₁ ^-1 ) AX .. (B9)

By the way, in the case of an operation modulo 2, the basic upper triangular matrix U _i (i = 1 to m) and the basic lower triangular matrix L _i (i = 1 to m) have the following properties.
U _i ≡ (U _i ) ^-1 (B10)
L _i ≡ (L _i ) ⁻¹ (B11)

Expression (B9) is transformed into the following expression by Expression (B10) and Expression (B11).
Y≡U ₁ U ₂ ... U _m-1 U _m L _m L _m-1 ... L ₂ L ₁ AX (B12)

Using the parity vector Y calculated by the equation (B12) and the information vector X, the encoded vector Z is expressed by the following equation.
Z≡ (X, Y) (B13)
Here, the first bit to the k-th bit of the encoded vector Z are bits of the information vector X, and the (k + 1) -th bit to the n-th bit of the encoded vector Z are bits of the parity vector Y.

(Constitution)
FIG. 2 is a diagram illustrating the configuration of the encoder according to the first embodiment.

  Referring to FIG. 2, this encoder 2a includes a parity check matrix decomposition unit 10, an LU decomposition unit 14, a basic matrix decomposition unit 16, an intermediate result storage unit 12, a matrix storage unit 18, and a matrix product calculation. Unit 11, array unit 20, and matrix product calculation control unit 121.

  The parity check matrix decomposition unit 10 decomposes the parity check matrix H into matrices A and B, stores data representing the matrix A in the matrix storage unit 18, and stores data representing the matrix B in the intermediate result storage unit 12. When the dimension of the information vector X is k, the dimension of the parity vector Y is m, and k + m = n, the check matrix H is an m × n matrix. The parity check matrix decomposition unit 10 extracts the first to kth columns from the parity check matrix H to form a matrix A, and extracts the (k + 1) th to nth columns to form a matrix B. A is an m × k matrix, and B is an m × m matrix.

  The LU decomposition unit 14 includes an adder 13 and a multiplier 15. The LU decomposition unit 14 performs LU decomposition on the matrix B according to the Crout algorithm to generate a lower triangular matrix L and an upper triangular matrix U.

The basic matrix decomposition unit 16 decomposes the upper triangular matrix U into a product of the basic upper triangular matrix U _i (i = 1 to m) as shown in Expression (B5). Further, the basic matrix decomposition unit 16 decomposes the lower triangular matrix L into a product of the basic lower triangular matrix L _i (i = 1 to m) as shown in Expression (B7).

In the basic upper triangular matrix U _i , the i-th row is the i-th row of the upper triangular matrix U, and in the rows other than the i-th row, the diagonal term element is 1 and the off-diagonal term element is 0. It is a matrix like this.
For example, when the upper triangular matrix U is represented by equation (1), the basic upper triangular matrices U ₁ , U ₂ , U ₃ are represented by equations (2), (3), and (4), respectively. It is.

In the basic lower triangular matrix L _i , the i-th column is the i-th column of the lower triangular matrix L, and in the columns other than the i-th column, the diagonal term element is 1 and the off-diagonal term element is 0. It is a certain matrix.
For example, when the lower triangular matrix L is represented by Expression (5), the basic lower triangular matrices L ₁ , L ₂ , and L ₃ are represented by Expression (6), Expression (7), and Expression (8), respectively. It is.

Matrix storage unit 18, the basic upper triangular matrix U _1, U _2, · · ·, data representing the U _m, the fundamental lower triangular matrix _{L m, L m-1,} ···, data representing the L _1, and matrix Data representing A is stored.

The matrix product calculation unit 11 includes an adder 17 and a multiplier 19. In order to calculate the parity vector Y, the matrix product calculation unit 11 performs (2m + 1) times of matrix multiplications included in the equation (B12) in the order instructed by the matrix product calculation control unit 121. Matrix product calculator 11, the information vector X is input, the matrix U _1, U ₂ in the matrix storage unit _{18, ···, U m, L} m, L m-1, ···, L 1, A Each matrix multiplication is executed using two of the data representing the intermediate data and the data representing the intermediate results in the intermediate result storage unit 12. The matrix product calculation unit 11 stores data representing the calculation result in the intermediate result storage unit 12 as data representing the intermediate result until (2m + 1) times of matrix multiplication included in the formula (B12) is completed.

Matrix product calculation control unit 121, in order to calculate the parity vector Y, A order from the head of the matrix data in the matrix storage unit 18 is _{stored, L 1, L 2, ···} , L m-1 , L _m , U _m , U _m−1 ,..., U ₂ , U ₁ are set in order, and the matrix product calculation unit 11 is set to perform matrix multiplication in this order. Control. Further, the matrix product calculation control unit 121 causes the matrix product calculation unit 11 to output a parity vector Y as a calculation result after the (2m + 1) times of matrix multiplication included in the formula (B12).

  The intermediate result storage unit 12 stores data representing an intermediate result in a process (matrix decomposition process) for generating a matrix included in the formula (B12). Further, the intermediate result storage unit 12 stores data representing an intermediate result of (2m + 1) times of matrix multiplication included in the equation (B12) during the generation process of the parity vector Y.

  The array unit 20 generates and outputs an n-bit encoded vector Z composed of the k-bit information vector X and the calculated m-bit parity vector Y.

(Matrix decomposition processing)
FIG. 3 is a flowchart of the matrix decomposition process according to the first embodiment.

  Referring to FIG. 3, parity check matrix decomposition unit 10 decomposes parity check matrix H into matrices A and B. The parity check matrix decomposition unit 10 stores data representing the matrix A in the matrix storage unit 18 and stores data representing the matrix B in the intermediate result storage unit 12 (step S101).

  Next, the LU decomposition unit 14 reads data representing the matrix B from the intermediate result storage unit 12, LU decomposes the matrix B, and obtains data representing the lower triangular matrix L and data representing the upper triangular matrix U. The result is stored in the intermediate result storage unit 12 (step S102).

Then, the base matrix decomposition unit 16, reads the data representative of the upper triangular matrix U from the intermediate result storage unit 12, the upper triangular matrix U base on triangular matrix U _1, U _2, · · ·, the U _m product The matrix storage unit 18 stores data representing the upper triangular matrix U ₁ , U ₂ ,..., U _m . Further, the base matrix decomposition unit 16 reads the data representing the lower triangular matrix L from the intermediate result storage unit 12, the basic lower triangular matrix the lower triangular matrix L L _1, L _2, · · ·, to the product of L _m By decomposing, data representing the basic lower triangular matrices L ₁ , L ₂ ,..., L _m is stored in the matrix storage unit 18 (step S103).

(Encoded vector generation processing)
FIG. 4 is a flowchart of the encoded vector generation process according to the first embodiment.

Referring to FIG. 4, matrix product calculation control section 121 determines that the order of the matrices in which data is stored in matrix storage section 18 is A, L ₁ , L ₂ ,..., L _m−1 , L from the top. The order between the matrices is set so as to be in the order of _m , U _m , U _m−1 ,..., U ₂ , U ₁ (step S201).

Next, the matrix product calculation control unit 121 sets the control variable i to 1 (step S202).
The matrix product calculation control unit 121 instructs the matrix product calculation unit 11 to read out the data representing the i-th (= 1) -th rank matrix A from the matrix storage unit 18, and the matrix A and the input information The product AX with the vector X is calculated. The matrix product calculation unit 11 sets the vector that is the multiplication result as the intermediate result R, and stores the data representing the intermediate result R in the intermediate result storage unit 12 (step S203).

  Next, the matrix product calculation control unit 121 increments the control variable i (step S204).

When the control variable i is less than (2m + 1) (YES in step S205), the matrix product calculation control unit 121 instructs the matrix product calculation unit 11 to read the matrix of the i-th rank from the matrix storage unit 18 causes read data representing the Mat _i), the intermediate result storage section 12 intermediate results are stored in the by reading the data representing the R, matrices Mat _i and an intermediate result product Mat _i × R and R Let's calculate. The matrix product calculation unit 11 sets the vector that is the multiplication result as a new intermediate result R, and updates the data representing the intermediate result R in the intermediate result storage unit 12 (step S206). Thereafter, the processing from step S204 is repeated.

When the control variable i becomes (2m + 1) (NO in step S205), the matrix product calculation control unit 121 instructs the matrix product calculation unit 11 to receive the i-th rank matrix (Mat) from the matrix storage unit 18. _i ) and the data representing the intermediate result R stored in the intermediate result storage unit 12 are read, and the product Mat _i × R of the matrix Mat _i and the intermediate result R is calculated. . The matrix product calculation unit 11 outputs the vector that is the multiplication result as the parity vector Y (step S207).

  Thereafter, the array unit 20 generates and outputs an n-bit encoded vector Z composed of a k-bit information vector X and an m-bit parity vector Y (step S208).

(Effects of the first embodiment)
As described above, according to the first embodiment, the parity vector Y is calculated according to the equation (B12) without performing backward substitution during the encoded vector generation process. Encoding can be performed. In particular, the basic upper triangular matrix U _i (i = 1 to m) and the basic lower triangular matrix L _i (i = 1 to m) are matrices with a small number of non-zero elements, so that the calculation amount of the expression (B12) is small. Few. Further, since the parity vector Y is calculated by sequentially executing the matrix multiplication on the right side of the equation (B12), the result of each matrix multiplication becomes a vector, and the capacity for storing the intermediate result can be reduced.

  Further, at the time of matrix decomposition processing, without performing an operation with a large amount of calculation of calculating an inverse matrix of the matrix B, LU decomposition of the matrix B, decomposition of the upper triangular matrix subjected to LU decomposition into a product of the basic upper triangular matrix Since it is only necessary to decompose the LU-decomposed lower triangular matrix into the product of the basic lower triangular matrix, a matrix necessary for generating the parity vector Y can be calculated with a small amount of calculation.

[Modification 1 of the first embodiment]
The encoder of the first modification of the first embodiment performs a real number operation.

(Encoding algorithm)
The encoder receives an information vector X composed of k information bits x1, x2,..., Xk and performs a real number operation to thereby obtain m parity bits y1, y2 satisfying the following expression. ,..., Ym to generate a parity vector Y.
H (X, Y) ^T = 0 (C1)
Here, H is a parity check matrix (hereinafter also referred to as a check matrix), which is a matrix of m rows and n columns (= k + m).
(X, Y) ^T represents transposition of a vector composed of an information vector and a parity vector.

Expression (C1) is modified as follows in the case of real number calculation.
BY = −AX (C2)
Here, the matrix A is a matrix of m rows and k columns including the first to kth columns of the check matrix H. The matrix B is a matrix of m rows and m columns including the (k + 1) -th column to the n-th column of the parity check matrix H.

When the matrix B is subjected to LU decomposition, the expression (C2) is transformed into the following expression.
LUY = −AX (C3)
Here, L is a lower triangular matrix and U is an upper triangular matrix.
From the equation (C3), the column vector Y is expressed by the following equation.
Y = −U ⁻¹ L ⁻¹ AX (C4)

The upper triangular matrix U is decomposed into products of basic upper triangular matrices U _i (i = 1 to m) as in the following equation.
U = U _m U _m-1 ... U ₂ U ₁ (C5)
From the equation (C5), the inverse matrix U ⁻¹ is represented by the following equation.
U ⁻¹ = U ₁ ⁻¹ U ₂ ⁻¹ ... U _m-1 ⁻¹ U _m ⁻¹ (C6)
Here, U _i ⁻¹ is obtained by inverting the sign of the element of U _{i in which} the element of the diagonal term is 1 and the element of the off-diagonal term.

The lower triangular matrix L is decomposed into products of basic lower triangular matrices L _i (i = 1 to m) as shown in the following expression.
L = L ₁ L ₂ ... L _m-1 L _m (C7)
From the equation (C7), the inverse matrix L ⁻¹ is represented by the following equation.
L ⁻¹ = L _m ⁻¹ L _m−1 ⁻¹ ... L ₂ ⁻¹ L ₁ ⁻¹ ... (C8)
Here, L _i ⁻¹ is obtained by inverting the sign of the element of L _{i in which} the element of the diagonal term is 1 and the element of the non-diagonal term.

Expression (C4) is transformed into the following expression by Expression (C6) and Expression (C8).
Y = -U ₁ ^-1 U ₂ ^-1 ... U _m-1 ^-1 U _m ^-1 L _m ^-1 L _m-1 ^-1 ... L ₂ ^-1 L ₁ ^-1 AX ( C9)

Using the parity vector Y calculated by the equation (C9) and the information vector X, the encoded vector Z is expressed by the following equation.
Z = (X, Y) (C10)
Here, the first bit to the k-th bit of the encoded vector Z are bits of the information vector X, and the (k + 1) -th bit to the n-th bit of the encoded vector Z are bits of the parity vector Y.

(Constitution)
FIG. 5 is a diagram illustrating a configuration of an encoder according to Modification 1 of the first embodiment.

  Referring to FIG. 5, the encoder 2b includes a parity check matrix decomposition unit 10, an LU decomposition unit 14, a basic matrix decomposition unit 16, and an arrangement unit 20, which are the same as those in the first embodiment. Description of these will not be repeated.

  The encoder 2 b includes a matrix storage unit 28, an inverse matrix calculation unit 23, a matrix product calculation unit 21, an intermediate result storage unit 22, a sign inversion unit 99, and a matrix product calculation control unit 122. .

Inverse matrix calculation unit 23, the basic upper triangular matrix U _1, U ₂ from the intermediate result storage unit 22, · · ·, U _m and the basic lower triangular matrix L _1, L _2, · · ·, the data representing the L _m read out, matrices _{U 1, U 2, ···,} U m, L 1, L 2, ···, by reversing the sign of the unpaired corner section elements of L _m, matrices U _1, U ₂ ,..., U _m , L ₁ , L ₂ ,..., L _m inverse matrices U ₁ ⁻¹ , U ₂ ⁻¹ ,..., U _m ⁻¹ , L ₁ ⁻¹ , L ₂ ^{− 1} ,..., L _m ⁻¹ is calculated and the inverse matrices U ₁ ⁻¹ , U ₂ ⁻¹ ,..., U _m ⁻¹ , L ₁ ⁻¹ , L ₂ ⁻¹ ,. Data representing _m ⁻¹ is stored in the matrix storage unit 28.

For example, when the matrix U ₄ is represented by Expression (9), the inverse matrix U ₄ ⁻¹ is represented by Expression (10).

For example, when the matrix L ₄ is represented by Expression (11), the inverse matrix L ₄ ⁻¹ is represented by Expression (12).

The matrix storage unit 28 includes data representing the inverse matrices U ₁ ⁻¹ , U ₂ ⁻¹ ,..., U _m ⁻¹ of the basic upper triangular matrix, and the inverse matrices L _m ⁻¹ and L _{m− of the} basic lower triangular matrix. Stores data representing ₁ ⁻¹ ,..., L ₁ ⁻¹ and data representing the matrix A.

The matrix product calculation unit 21 includes an adder 27 and a multiplier 29. In order to calculate the parity vector Y, the matrix product calculation unit 21 performs (2m + 1) times of matrix multiplication included in the formula (C9) in the order instructed by the matrix product calculation control unit 122. Matrix product calculator 21, the information vector X is input, the matrix U ₁ ^-1 in the matrix storage unit ^{_{28, U 2 -1, ···,}} U m -1, L m -1, L m-1 - ¹ ,..., L ₁ ⁻¹ , data representing A, and data representing intermediate results in the intermediate result storage unit 22 are used to perform each matrix multiplication. The matrix product calculation unit 21 stores data representing the calculation result in the intermediate result storage unit 22 as data representing the intermediate result until (2m + 1) times of matrix multiplications included in the formula (C9) is completed.

In order to calculate the parity vector Y, the matrix product calculation control unit 122 determines that the order of the matrices in which data is stored in the matrix storage unit 28 is A, L ₁ ⁻¹ , L ₂ ⁻¹ ,. Set the order between the matrices so that L _m-1 ⁻¹ , L _m ⁻¹ , U _m ⁻¹ , U _m−1 ⁻¹ ,..., U ₂ ⁻¹ , U ₁ ⁻¹ are in this order. The matrix product calculation unit 21 is controlled to perform matrix multiplication in this order. Further, the matrix product calculation control unit 122, after the completion of (2m + 1) matrix multiplications included in the formula (C9), the vector Y ′ (= U ₁ ⁻¹ U ₂ ^{−) as a} calculation result from the matrix product calculation unit 21. ¹ ... U _m−1 ⁻¹ U _m ⁻¹ L _m ⁻¹ L _m−1 ⁻¹ ... L ₂ ⁻¹ L ₁ ⁻¹ AX) is output to the sign inversion unit 99.

  The intermediate result storage unit 22 stores data representing an intermediate result in a process (matrix decomposition process) for generating a matrix included in the formula (C9). Further, the intermediate result storage unit 22 stores data representing an intermediate result of (2m + 1) times of matrix multiplication included in the equation (C9) during the generation process of the parity vector Y.

  The sign inversion unit 99 inverts the sign of the vector Y ′ (that is, inverts the sign of each element of the vector) and outputs it as the parity vector Y.

(Matrix decomposition processing)
FIG. 6 is a flowchart of the matrix decomposition process according to Modification 1 of the first embodiment.

  Referring to FIG. 6, parity check matrix decomposition unit 10 decomposes parity check matrix H into matrices A and B. The parity check matrix decomposition unit 10 stores data representing the matrix A in the matrix storage unit 28, and stores data representing the matrix B in the intermediate result storage unit 22 (step S301).

  Next, the LU decomposition unit 14 reads the data representing the matrix B from the intermediate result storage unit 22, performs LU decomposition on the matrix B, and obtains data representing the lower triangular matrix L and data representing the upper triangular matrix U. The result is stored in the intermediate result storage unit 22 (step S302).

Then, the base matrix decomposition unit 16, reads the data representative of the upper triangular matrix U from the intermediate result storage unit 22, the upper triangular matrix U base on triangular matrix U _1, U _2, · · ·, the U _m product And the data representing the basic triangular matrix U ₁ , U ₂ ,..., U _m are stored in the intermediate result storage unit 22. Further, the base matrix decomposition unit 16 reads the data representing the lower triangular matrix L from the intermediate result storage unit 22, the basic lower triangular matrix the lower triangular matrix L L _1, L _2, · · ·, to the product of L _m decompose, basic lower triangular matrix L _1, L _2, · · ·, and stores the data representing the L _m in the intermediate result storage unit 22 (step S303).

Inverse matrix calculation unit 23, the basic upper triangular matrix U _1, U ₂ from the intermediate result storage unit 22, · · ·, U _m and the basic lower triangular matrix L _1, L _2, · · ·, the data representing the L _m read out, matrices _{U 1, U 2, ···,} U m, L 1, L 2, ···, by reversing the sign of the unpaired corner section elements of L _m, matrices U _1, U ₂ ,..., U _m , L ₁ , L ₂ ,..., L _m inverse matrices U ₁ ⁻¹ , U ₂ ⁻¹ ,..., U _m ⁻¹ , L ₁ ⁻¹ , L ₂ ^{− 1} ,..., L _m ⁻¹ is calculated and the inverse matrices U ₁ ⁻¹ , U ₂ ⁻¹ ,..., U _m ⁻¹ , L ₁ ⁻¹ , L ₂ ⁻¹ ,. Data representing _m ⁻¹ is stored in the matrix storage unit 28 (step S304).

(Encoded vector generation processing)
FIG. 7 is a flowchart of an encoded vector generation process according to Modification 1 of the first embodiment.

Referring to FIG. 7, the matrix product calculation control unit 122 determines that the order of the matrices in which data is stored in the matrix storage unit 28 is A, L ₁ ⁻¹ , L ₂ ⁻¹ _{,. −1} ⁻¹ , L _m ⁻¹ , U _m ⁻¹ , U _m−1 ⁻¹ ,..., U ₂ ⁻¹ , U ₁ ⁻¹ are set in an order between the matrices (step S401).

Next, the matrix product calculation control unit 122 sets the control variable i to 1 (step S402).
The matrix product calculation control unit 122 instructs the matrix product calculation unit 21 to read out the data representing the i-th (= 1) -th rank matrix A from the matrix storage unit 28, and the matrix A and the input information The product AX with the vector X is calculated. The matrix product calculation unit 21 sets the vector that is the multiplication result as the intermediate result R, and stores the data representing the intermediate result R in the intermediate result storage unit 22 (step S403).

  Next, the matrix product calculation control unit 122 increments the control variable i (step S404).

If the control variable i is less than (2m + 1) (YES in step S405), the matrix product calculation control unit 122 instructs the matrix product calculation unit 21 to read the i-th rank matrix from the matrix storage unit 28. causes read data representing the Mat _i), the intermediate result storage section 22 intermediate results are stored in the by reading the data representing the R, matrices Mat _i and an intermediate result product Mat _i × R and R Let's calculate. The matrix product calculation unit 21 sets the vector that is the multiplication result as a new intermediate result R, and updates data representing the intermediate result R in the intermediate result storage unit 22 (step S406). Thereafter, the processing from step S404 is repeated.

When the control variable i becomes (2m + 1) (NO in step S405), the matrix product calculation control unit 122 instructs the matrix product calculation unit 21 to receive the i-th rank matrix (Mat) from the matrix storage unit 28. _i ) and data representing the intermediate result R stored in the intermediate result storage unit 22 are read, and a product Mat _i × R of the matrix Mat _i and the intermediate result R is calculated. . The matrix product calculation unit 21 outputs a vector Y ′ as a multiplication result to the sign inversion unit 99. The sign reversing unit 99 outputs the inverted vector Y ′ as the parity vector Y (step S407).

  Thereafter, the arrangement unit 20 generates and outputs an n-bit encoded vector Z composed of a k-bit information vector X and an m-bit parity vector Y (step S408).

(Effects of Modification 1 of First Embodiment)
As described above, according to the first modification of the first embodiment, the parity vector Y is calculated according to the equation (C9) without performing backward substitution at the time of the encoded vector generation process. The information vector X can be encoded. In particular, the inverse matrix U _i ⁻¹ (i = ^{1 to} m) of the basic upper triangular matrix and the inverse matrix L _i ⁻¹ (i = ^{1 to} m) of the basic lower triangular matrix are matrices with a small number of nonzero elements. There is little calculation amount of the calculation of Formula (C9). Further, since the parity vector Y is calculated by sequentially executing from the matrix multiplication on the right side of the equation (C9), the result of each matrix multiplication becomes a vector, and the capacity for storing the intermediate result can be reduced.

  Further, at the time of matrix decomposition processing, without performing an operation with a large amount of calculation of calculating an inverse matrix of the matrix B, LU decomposition of the matrix B, decomposition of the upper triangular matrix subjected to LU decomposition into a product of the basic upper triangular matrix Since the decomposition of the LU-decomposed lower triangular matrix into the product of the basic lower triangular matrix and the inversion of the signs of the non-diagonal terms of the basic upper triangular matrix and the basic lower triangular matrix, the parity vector Y can be performed with a small amount of calculation. The matrix required to generate can be calculated.

[Second Embodiment]
The encoder of the second embodiment performs an operation modulo 2.

(Encoding algorithm)
The encoder receives an information vector X consisting of k information bits x1, x2,..., Xk and performs an operation modulo 2 so as to satisfy m A parity vector Y composed of bits y1, y2,.
H (X, Y) ^T ≡0 (D1)
Here, H is a parity check matrix (hereinafter also referred to as a check matrix), which is a matrix of m rows and n columns (= k + m).
(X, Y) ^T represents transposition of a vector composed of an information vector and a parity vector.

Expression (D1) is modified as follows in the case of an operation modulo 2.
BY≡AX (D2)
Here, the matrix A is a matrix of m rows and k columns including the first to kth columns of the check matrix H. The matrix B is a matrix of m rows and m columns including the (k + 1) -th column to the n-th column of the parity check matrix H.

Assume that ordering is performed on the matrix B as follows.
First, when two rows of the matrix B are exchanged, the equation (D2) is transformed as follows.
P ₁ BY≡P ₁ AX (D3)
Here, P ₁ is a transformation matrix representing the first row replacement. In the case of converting i row and j row, P ₁ is a matrix in which the i row and j row of the unit matrix E are exchanged.

Furthermore, when the two columns of the matrix B are exchanged, the equation (D3) is transformed as follows.
P ₁ BQ ₁ Q ₁ ^-1 Y≡P ₁ AX (D4)
Here, Q ₁ is a transformation matrix representing the first column replacement. When converting the i column and the j column, Q ₁ is a matrix in which the i column and the j column of the unit matrix E are exchanged.

Define a new matrix as follows:
B ⁽¹⁾ ≡ P ₁ BQ ₁ (D5)
A ⁽¹⁾ ≡ P ₁ A (D6)
From formula (D5) and formula (D6), formula (D4) is represented by the following formula.
B ⁽¹⁾ Q ₁ ^-1 Y≡A ⁽¹⁾ X (D7)

Further, when two rows of the matrix B ⁽¹⁾ are exchanged, the equation (D7) is transformed as follows.
P ₂ B ⁽¹⁾ Q ₁ ⁻¹ Y≡P ₂ A ⁽¹⁾ X (D8)
Here, P ₂ is a transformation matrix representing the second row replacement.

Further, when the two columns of the matrix B ⁽¹⁾ are exchanged, the equation (D8) is transformed as follows.
P ₂ B ⁽¹⁾ Q ₂ Q ₂ ^-1 Q ₁ ^-1 Y≡P ₂ A ⁽¹⁾ X (D9)
Here, Q ₂ is a transformation matrix representing the second column replacement.

Define a new matrix as follows:
B ⁽²⁾ ≡ P ₂ B ⁽¹⁾ Q ₂ (D10)
A ⁽²⁾ ≡ P ₂ A ⁽¹⁾ (D11)
From formula (D10) and formula (D11), formula (D9) is represented by the following formula.
B ⁽²⁾ Q ₂ ^-1 Q ₁ ^-1 Y≡A ⁽²⁾ X (D12)

Similarly, for the matrix B, the row replacement and the v-th (v ≧ 1) extra column are performed for the u-th (u ≧ 1). Assuming that the matrix representing the s-th (1 ≦ s ≦ u) row replacement is P _s and the matrix representing the s-th (1 ≦ s ≦ v) column replacement is Q _s , the following new matrix Define
B′≡P _u ... P ₂ P ₁ BQ ₁ Q ₂ ... Q _v (D13)
A ^(u) ≡P _u ... P ₂ P ₁ A (D14)
E ^(v) ≡Q ₁ Q ₂ ... Q _v ... (D15)
The following expressions are established by Expression (D13), Expression (D14), and Expression (D15).
B′E ^{(v) -1} Y≡A ^(u) X (D16)

When the matrix B ′ is subjected to LU decomposition, the equation (D16) is transformed into the following equation.
L′ U′E ^{(v) -1} Y≡A ^(u) X (D17)
Here, L ′ is a lower triangular matrix and U ′ is an upper triangular matrix.

From the equation (D17), the column vector Y is expressed by the following equation.
Y≡E ^(v) U ′ ⁻¹ L ′ ⁻¹ A ^(u) X (D18)

The upper triangular matrix U ′ is decomposed into products of basic upper triangular matrices U ′ _i (i = 1 to m) as in the following equation.
_{U'≡U 'm U' m-1} ··· U '2 U' 1 ··· (D19)
From the equation (D19), the inverse matrix U ′ ⁻¹ is expressed by the following equation.
U ' ^-1 ≡ U' ₁ ^-1 U ' ₂ ^-1 ... U' _m-1 ^-1 U ' _m ^-1 ... (D20)

The lower triangular matrix L ′ is decomposed into products of basic lower triangular matrices L ′ _i (i = 1 to m) as in the following equation.
L′ ≡L ′ ₁ L ′ ₂ ... L ′ _m−1 L ′ _m (D21)
From the equation (D21), the inverse matrix L ′ ⁻¹ is expressed by the following equation.
L ′ ⁻¹ ≡L ′ _m ⁻¹ L ′ _m−1 ⁻¹ ... L ′ ₂ ⁻¹ L ′ ₁ ⁻¹ (D22)

Expression (D18) is transformed into the following expression by Expression (D20) and Expression (D22).
Y≡E ^(v) (U ′ ₁ ⁻¹ U ′ ₂ ⁻¹ ... U ′ _m-1 ⁻¹ U ′ _m ⁻¹ ) (L ′ _m ⁻¹ L ′ _m−1 ⁻¹ ... L ' ₂ ^-1 L' ₁ ^-1 ) A ^(u) X (D23)

By the way, in the case of operation modulo 2, the basic upper triangular matrix U ′ _i (i = 1 to m) and the basic lower triangular matrix L ′ _i (i = 1 to m) have the following properties. Have.
U ′ _i ≡ (U ′ _i ) ⁻¹ (D24)
L ′ _i ≡ (L ′ _i ) ⁻¹ (D25)
Expression (D23) is transformed into the following expression by Expression (D24) and Expression (D25).
Y≡E ^(v) U ' ₁ U' ₂ ... U ' _m-1 U' _m L ' _m L' _m-1 ... L ' ₂ L' ₁ A ^(u) X (D26 )

Using the parity vector Y calculated by the equation (D26) and the information vector X, the encoded vector Z is expressed by the following equation.
Z≡ (X, Y) (D27)
Here, the first bit to the k-th bit of the encoded vector Z are bits of the information vector X, and the (k + 1) -th bit to the n-th bit of the encoded vector Z are bits of the parity vector Y.

(Constitution)
FIG. 8 is a diagram illustrating the configuration of the encoder according to the second embodiment.

  Referring to FIG. 8, the encoder 2c includes a parity check matrix decomposition unit 30, an ordering unit 40, an LU decomposition unit 34, a basic matrix decomposition unit 36, an intermediate result storage unit 32, and a matrix storage unit 38. A matrix product calculation unit 31, an array unit 20, and a matrix product calculation control unit 123.

  The parity check matrix decomposition unit 30 decomposes the parity check matrix H into matrices A and B, and causes the intermediate result storage unit 32 to store data representing the matrix A and data representing the matrix B.

The ordering unit 40 performs u-th (u ≧ 1) row replacement and v-th time with respect to the matrix B so that the number of non-zero elements of the upper triangular matrix and the lower triangular matrix obtained by LU decomposition is reduced. The matrix B ′ is generated by exchanging columns (v ≧ 1). As an ordering method, for example, a minimum order method can be used. Further, the ordering unit 40 performs the same replacement as the u-time replacement of the matrix B with respect to the matrix A to generate the matrix A ^(u), and generates v times for the matrix B with respect to the unit matrix E. The matrix E ^(v) is generated by performing the same replacement as the replacement of the columns.

  The LU decomposition unit 34 includes an adder 33 and a multiplier 35. The LU decomposition unit 34 performs LU decomposition on the matrix B ′ according to the Crouto algorithm to generate a lower triangular matrix L ′ and an upper triangular matrix U ′.

The basic matrix decomposition unit 36 decomposes the upper triangular matrix U ′ into a product of the basic upper triangular matrix U ′ _i (i = 1 to m) as shown in Expression (D19). Further, the basic matrix decomposition unit 36 decomposes the lower triangular matrix L ′ into a product of the basic lower triangular matrix L ′ _i (i = 1 to m) as shown in Expression (D21).

Matrix storage unit 38, the basic upper triangular matrix _{U '1, U' 2,} ···, U ' data representing the _m, basic lower triangular matrix _{L' m, L 'm-} 1, ···, L' 1 And data representing the matrices A ^(u) and E ^(v) are stored.

The matrix product calculation unit 31 includes an adder 37 and a multiplier 39. In order to calculate the parity vector Y, the matrix product calculation unit 31 performs (2m + 2) times of matrix multiplications included in the equation (D26) in the order instructed by the matrix product calculation control unit 123. The matrix product calculation unit 31 receives the input information vector X, the matrices U ′ ₁ , U ′ ₂ ,..., U ′ _m , L ′ _m , L ′ _m−1 ,. , L ′ ₁ , A ^(u) , E ^(v) , and data representing the intermediate results in the intermediate result storage unit 32 are used to perform each matrix multiplication. The matrix product calculation unit 31 stores data representing the calculation result in the intermediate result storage unit 32 as data representing the intermediate result until (2m + 2) times of matrix multiplications included in the equation (D26) is completed.

In order to calculate the parity vector Y, the matrix product calculation control unit 123 sets the order of the matrices in which the data is stored in the matrix storage unit 38 from the head A ^(u) , L ′ ₁ , L ′ ₂ ,. , L ′ _m , U ′ _m , U ′ _m−1 ,..., U ′ ₁ , E ^(v) are set in order, and matrix multiplication is performed in this order. The matrix product calculation unit 31 is controlled. In addition, the matrix product calculation control unit 123 causes the matrix product calculation unit 31 to output the parity vector Y as a calculation result after the (2m + 2) times of matrix multiplication included in the equation (D26).

  The intermediate result storage unit 32 stores data representing an intermediate result in a process (matrix decomposition process) for generating a matrix included in the equation (D26). Further, the intermediate result storage unit 32 stores data representing an intermediate result of (2m + 2) times of matrix multiplication included in the equation (D26) during the generation process of the parity vector Y.

  The array unit 20 generates and outputs an n-bit encoded vector Z composed of the k-bit information vector X and the calculated m-bit parity vector Y.

(Matrix decomposition processing)
FIG. 9 is a flowchart of matrix decomposition processing according to the second embodiment.

  Referring to FIG. 9, parity check matrix decomposition unit 30 decomposes parity check matrix H into matrices A and B. The parity check matrix decomposition unit 30 stores data representing the matrix A and data representing the matrix B in the intermediate result storage unit 32 (step S501).

The ordering unit 40 performs u-th (u ≧ 1) row replacement and v-th (v ≧ 1) column replacement for the matrix B. The ordering unit 40 uses P _{s as} a matrix representing the s-th (1 ≦ s ≦ u) row replacement, Q _s as a matrix representing the s-th (1 ≦ s ≦ v) column replacement, and A new matrix is generated as follows.
B′≡P _u ... P ₂ P ₁ BQ ₁ Q ₂ ... Q _v (D13)
A ^(u) ≡P _u ... P ₂ P ₁ A (D14)
E ^(v) ≡Q ₁ Q ₂ ... Q _v ... (D15)
That is, the matrix B ′ is obtained by replacing the matrix B with u times (u ≧ 1) and v times (v ≧ 1). The matrix A ^(u) is obtained by performing the same replacement on the matrix A as the u-time replacement of rows for the matrix B. The matrix E ^(v) is obtained by performing the same replacement as the unit matrix E with v column replacements for the matrix B. The ordering unit 40 stores data representing the matrices A ^(u) and E ^(v) in the matrix storage unit 38 and stores data representing the matrix B ′ in the intermediate result storage unit 32 (step S502).

  Next, the LU decomposition unit 34 reads the data representing the matrix B ′ from the intermediate result storage unit 32, performs LU decomposition on the matrix B ′, and converts the data representing the lower triangular matrix L ′ and the upper triangular matrix U ′. The data to be represented is stored in the intermediate result storage unit 32 (step S503).

Next, the basic matrix decomposition unit 36 reads data representing the upper triangular matrix U ′ from the intermediate result storage unit 32, and converts the upper triangular matrix U ′ into the basic upper triangular matrices U ′ ₁ , U ′ ₂ ,. 'is decomposed into a product of _m, the basic upper triangular matrix _{U' U 1, U '2} , ···, U' and stores the data representing the _m in the matrix storage unit 38. The basic matrix decomposition unit 36 reads data representing the lower triangular matrix L ′ from the intermediate result storage unit 32, and converts the lower triangular matrix L into the basic lower triangular matrices L ′ ₁ , L ′ ₂ ,. _The matrix storage unit 38 stores the data representing the basic lower triangular matrices L ′ ₁ , L ′ ₂ ,..., L ′ _m by dividing into _m products (step S504).

(Encoded vector generation processing)
FIG. 10 is a flowchart of the encoded vector generation process according to the second embodiment.

Referring to FIG. 10, the matrix product calculation control unit 123 determines that the order of the matrices in which data is stored in the matrix storage unit 38 is A ^(u) , L ′ ₁ , L ′ ₂ ,. _{'m, U' m, U} 'm-1, ···, U' 1, sets the order between the matrix such that the order of E ^(v) (step S601).

Next, the matrix product calculation control unit 123 sets the control variable i to 1 (step S602).
The matrix product calculation control unit 123 instructs the matrix product calculation unit 31 to read out the data representing the i th (= 1) -th order matrix A ^(u) from the matrix storage unit 38, and the matrix A ^(u) And the product A ^(u) X of the input information vector X is calculated. The matrix product calculation unit 31 sets the vector that is the multiplication result as the intermediate result R, and stores the data representing the intermediate result R in the intermediate result storage unit 32 (step S603).

  Next, the matrix product calculation control unit 123 increments the control variable i (step S604).

When the control variable i is less than (2m + 2) (YES in step S605), the matrix product calculation control unit 123 instructs the matrix product calculation unit 31 to read the i-th rank matrix (the matrix) from the matrix storage unit 38. causes read data representing the Mat _i), the intermediate result storage unit 32 intermediate results are stored in the by reading the data representing the R, matrices Mat _i and an intermediate result product Mat _i × R and R Let's calculate. The matrix product calculation unit 31 sets the vector that is the multiplication result as a new intermediate result R, and updates the data representing the intermediate result R in the intermediate result storage unit 32 (step S606). Thereafter, the processing from step S604 is repeated.

When the control variable i becomes (2m + 2) (NO in step S605), the matrix product calculation control unit 123 instructs the matrix product calculation unit 31 to receive the i-th rank matrix (Mat) from the matrix storage unit 38. _i ) and the data representing the intermediate result R stored in the intermediate result storage unit 32 are read, and the product Mat _i × R of the matrix Mat _i and the intermediate result R is calculated. . The matrix product calculation unit 31 outputs the vector that is the multiplication result as the parity vector Y (step S607).

  Thereafter, the array unit 20 generates and outputs an n-bit encoded vector Z composed of a k-bit information vector X and an m-bit parity vector Y (step S608).

(Effect of 2nd Embodiment)
As described above, according to the second embodiment, since the parity vector Y is calculated according to the equation (D26) without performing backward substitution in the encoded vector generation process, the information vector X can be calculated with a small amount of calculation. Encoding can be performed. In particular, since the matrix B is ordered, the basic upper triangular matrix U ′ _i (i = 1 to m) and the basic lower triangular matrix L ′ _i (i = 1 to m) are the basic upper triangles of the first embodiment. The matrix U _i (i = 1 to m) and the basic lower triangular matrix L _i (i = 1 to m) have a smaller number of non-zero elements, and the calculation amount of the expression (D26) is expressed by the expression (B12). Can be even less. Further, since the parity vector Y is calculated by sequentially executing the matrix multiplication on the right side of the equation (D26), the result of each matrix multiplication becomes a vector, and the capacity for storing the intermediate result can be reduced.

Further, at the time of matrix decomposition processing, without performing an operation with a large amount of calculation of calculating an inverse matrix of the matrix B, generation of matrices B ′, A ^(u) and E ^(v) by ordering of the matrix B, matrix B LU decomposition, LU decomposition of upper triangular matrix into basic upper triangular matrix product, and LU decomposition of lower triangular matrix into basic lower triangular matrix product. A matrix necessary for generating the parity vector Y can be calculated.

[Modification 1 of Second Embodiment]
The encoder according to the first modification of the second embodiment performs a real number operation.

(Encoding algorithm)
The encoder receives an information vector X composed of k information bits x1, x2,..., Xk and performs a real number operation to thereby obtain m parity bits y1, y2 satisfying the following expression. ,..., Ym to generate a parity vector Y.
H (X, Y) ^T = 0 (E1)
Here, H is a parity check matrix (hereinafter also referred to as a check matrix), which is a matrix of m rows and n columns (= k + m).
(X, Y) ^T represents transposition of a vector composed of an information vector and a parity vector.

Expression (E1) is modified as follows in the case of real number calculation.
BY = −AX (E2)
Here, the matrix A is a matrix of m rows and k columns including the first to kth columns of the check matrix H. The matrix B is a matrix of m rows and m columns including the (k + 1) -th column to the n-th column of the parity check matrix H.

Assume that ordering is performed on the matrix B as follows.
First, when two rows of the matrix B are exchanged, the equation (E2) is transformed as follows.
P ₁ BY = P ₁ AX (E3)
Here, P ₁ is a transformation matrix representing the first row replacement. In the case of converting i row and j row, P ₁ is a matrix in which the i row and j row of the unit matrix E are exchanged.

Further, when the two columns of the matrix B are exchanged, the equation (E3) is transformed as follows.
P ₁ BQ ₁ Q ₁ ^-1 Y = -P ₁ AX (E4)
Here, Q ₁ is a transformation matrix representing the first column replacement. When converting the i column and the j column, Q ₁ is a matrix in which the i column and the j column of the unit matrix E are exchanged.

Define a new matrix as follows:
B ⁽¹⁾ = P ₁ BQ ₁ (E5)
A ⁽¹⁾ = P ₁ A (E6)
From the equations (E5) and (E6), the equation (E4) is represented by the following equation.
B ⁽¹⁾ Q ₁ ⁻¹ Y = −A ⁽¹⁾ X (E7)

Further, if two rows of the matrix B ⁽¹⁾ are exchanged, the equation (E7) is transformed as follows.
P ₂ B ⁽¹⁾ Q ₁ ⁻¹ Y = −P ₂ A ⁽¹⁾ X (E8)
Here, P ₂ is a transformation matrix representing the second row replacement.

Further, when the two columns of the matrix B ⁽¹⁾ are exchanged, the equation (E8) is transformed as follows.
P ₂ B ⁽¹⁾ Q ₂ Q ₂ ^-1 Q ₁ ^-1 Y = -P ₂ A ⁽¹⁾ X (E9)
Here, Q ₂ is a transformation matrix representing the second column replacement.

Define a new matrix as follows:
B ⁽²⁾ = P ₂ B ⁽¹⁾ Q ₂ (E10)
A ⁽²⁾ = P ₂ A ⁽¹⁾ ... (E11)
From the equations (E10) and (E11), the equation (E9) is represented by the following equation.
B ⁽²⁾ Q ₂ ^-1 Q ₁ ^-1 Y = -A ⁽²⁾ X (E12)

Similarly, for the matrix B, the row replacement and the v-th (v ≧ 1) extra column are performed for the u-th (u ≧ 1). Assuming that the matrix representing the s-th (1 ≦ s ≦ u) row replacement is P _s and the matrix representing the s-th (1 ≦ s ≦ v) column replacement is Q _s , the following new matrix Define
_{B '= P u ··· P 2} P 1 BQ 1 Q 2 ··· Q v ··· (E13)
A ^(u) = P _u ... P ₂ P ₁ A (E14)
E ^(v) = Q ₁ Q ₂ ... Q _v (E15)
The following formulas are established by formula (E13), formula (E14), and formula (E15).
B'E ^{(v) -1} Y = -A ^(u) X (E16)

When the matrix B ′ is subjected to LU decomposition, the equation (E16) is transformed into the following equation.
L′ U′Q _V ⁻¹ ... Q ₂ ⁻¹ Q ₁ ⁻¹ Y = A ^(u) X (E17)
Here, L ′ is a lower triangular matrix and U ′ is an upper triangular matrix.

From the equation (E17), the column vector Y is expressed by the following equation.
Y = −E ^(v) U ′ ⁻¹ L ′ ⁻¹ A ^(u) X (E18)

The upper triangular matrix U ′ is decomposed into products of basic upper triangular matrices U ′ _i (i = 1 to m) as in the following equation.
_{U '= U' m U '} m-1 ··· U' 2 U '1 ··· (E19)
Here, in the inverse matrix U ′ _i ⁻¹ , the element of the diagonal term is 1, and the element of the off-diagonal term is obtained by inverting the sign of the element of U ′ _i .
From the equation (E19), U ′ ⁻¹ is represented by the following equation.
U ′ ⁻¹ = U ′ ₁ ⁻¹ U ′ ₂ ⁻¹ ... U ′ _m−1 ⁻¹ U ′ _m ⁻¹ (E20)

The lower triangular matrix L is decomposed into products of basic lower triangular matrices L _i (i = 1 to m) as shown in the following expression.
L ′ = L ′ ₁ L ′ ₂ ... L ′ _m−1 L ′ _m (E21)
Here, in the inverse matrix L ′ _i ⁻¹ , the element of the diagonal term is 1, and the element of the non-diagonal term is obtained by inverting the sign of the element of L ′ _i .
From the formula (E21), L ′ ⁻¹ is represented by the following formula.
L ′ ⁻¹ = L ′ _m ⁻¹ L ′ _m−1 ⁻¹ ... L ′ ₂ ⁻¹ L ′ ₁ ⁻¹ (E22)

Expression (E18) is transformed into the following expression by Expression (E20) and Expression (E22).
Y = -E ^(v) U ' ₁ ^-1 U' ₂ ^-1 ... U ' _m-1 ^-1 U' _m ^-1 L ' _m ^-1 L' _m-1 ^-1 ... L ' ₂ ^-1 L ' ₁ ^-1 A ^(u) X (E23)

Using the parity vector Y calculated by the equation (E23) and the information vector X, the encoded vector Z is expressed by the following equation.
Z = (X, Y) (E24)
Here, the first bit to the k-th bit of the encoded vector Z are bits of the information vector X, and the (k + 1) -th bit to the n-th bit of the encoded vector Z are bits of the parity vector Y.

(Constitution)
FIG. 11 is a diagram illustrating a configuration of an encoder according to Modification 1 of the second embodiment.

  Referring to FIG. 11, this encoder 2d includes a parity check matrix decomposition unit 30, an ordering unit 40, an LU decomposition unit 34, a basic matrix decomposition unit 36, and an array unit, which are the same as those in the second embodiment. 20. Description of these will not be repeated.

  The encoder 2d includes a matrix storage unit 48, an inverse matrix calculation unit 43, a matrix product calculation unit 41, an intermediate result storage unit 42, a sign inversion unit 99, and a matrix product calculation control unit 124. .

The inverse matrix calculator 43 stores the basic upper triangular matrices U ′ ₁ , U ′ ₂ ,..., U ′ _m and the basic lower triangular matrices L ′ ₁ , L ′ ₂ ,. 'reads the data representing the _m, the matrix _{U' 1, U '2,} ···, U' m, L '1, L' 2, ···, unpaired corner section elements of L _'m code by inverting the matrix _{U '1, U' 2,} ···, U 'm, L' 1, L '2, ···, L' inverse matrix U of _{m '1} ^-1, U' _{2 ^{-1, ···, U 'm -1}} , L' 1 -1, L '2 -1, ···, L' calculates _m ^-1, the inverse matrix ^{_{U '1 -1, U' 2}} - ^{_{1, ···, U 'm -1}} , L' 1 -1, L '2 -1, ···, L' and stores the data representing the _m ^-1 in the matrix storage unit 48.

The matrix storage unit 48 includes data representing the inverse matrices U ′ ₁ ⁻¹ , U ′ ₂ ⁻¹ ,..., U ′ _m ⁻¹ of the basic upper triangular matrix, and the inverse matrix L ′ _m ^{−1 of the} basic lower triangular matrix , L ′ _m−1 ⁻¹ ,..., L ′ ₁ ⁻¹ and data representing the matrices A ^(u) and E ^(v) are stored.

The matrix product calculation unit 41 includes an adder 47 and a multiplier 49. In order to calculate the parity vector Y, the matrix product calculation unit 41 performs (2m + 2) times of matrix multiplications included in the equation (E23) in the order instructed by the matrix product calculation control unit 124. The matrix product calculation unit 41 inputs the information vector X, the matrices U ′ ₁ ⁻¹ , U ′ ₂ ⁻¹ ,..., U ′ _m ⁻¹ , L ′ _m ⁻¹ , L in the matrix storage unit 48 ′ M ₋₁ ⁻¹ ,..., L ′ ₁ ⁻¹ , data representing A ^(u) and E ^(v) , and data representing intermediate results in the intermediate result storage unit 42 are used. To perform each matrix multiplication. The matrix product calculation unit 41 stores data representing the calculation result in the intermediate result storage unit 42 as data representing the intermediate result until (2m + 2) times of matrix multiplications included in the equation (E23) is completed.

In order to calculate the parity vector Y, the matrix product calculation control unit 124 determines that the order of the matrices in which data is stored in the matrix storage unit 48 is A ^(u) , L ′ ₁ ⁻¹ , L ′ ₂ ^{−1 from the top.} ,..., L ′ _m−1 ⁻¹ , L ′ _m ⁻¹ , U ′ _m ⁻¹ , U ′ _m−1 ⁻¹ ,..., U ′ ₂ ⁻¹ , U ′ ₁ ⁻¹ , E The order between the matrices is set so as to be in the order of ^(v) , and the matrix product calculation unit 41 is controlled so as to perform matrix multiplication in this order. Further, the matrix product calculation control unit 124, after the completion of (2m + 2) times of matrix multiplication included in the equation (E23), the vector Y ′ (= E ^(v) U ′ ₁ ^{) as the} calculation result from the matrix product calculation unit 41. ^-1 U ' ₂ ^-1 ... U' _m-1 ^-1 U ' _m ^-1 L' _m ^-1 L ' _m-1 ^-1 ... L' ₂ ^-1 L ' ₁ ^-1 A ^{(u )} ) Is output to the sign inversion unit 99.

  The intermediate result storage unit 42 stores data representing an intermediate result in a process (matrix decomposition process) for generating a matrix included in the equation (E23). Further, the intermediate result storage unit 42 stores data representing an intermediate result of (2m + 2) times of matrix multiplication included in the equation (E23) during the generation process of the parity vector Y.

  The sign inversion unit 99 inverts the sign of the vector Y ′ (that is, inverts the sign of each element of the vector) and outputs it as the parity vector Y.

(Matrix decomposition processing)
FIG. 12 is a flowchart of matrix decomposition processing according to the first modification of the second embodiment.

  Referring to FIG. 12, parity check matrix decomposition unit 30 decomposes parity check matrix H into matrices A and B. The parity check matrix decomposition unit 30 stores data representing the matrix A and data representing the matrix B in the intermediate result storage unit 42 (step S701).

The ordering unit 40 performs u-th (u ≧ 1) row replacement and v-th (v ≧ 1) column replacement for the matrix B. The ordering unit 40 uses P _{s as} a matrix representing the s-th (1 ≦ s ≦ u) row replacement, Q _s as a matrix representing the s-th (1 ≦ s ≦ v) column replacement, and A new matrix is generated as follows.
_{B '= P u ··· P 2} P 1 BQ 1 Q 2 ··· Q v ··· (D13)
A ^(u) = P _u ... P ₂ P ₁ A (D14)
E ^(v) = Q ₁ Q ₂ ... Q _v (D15)
That is, the matrix B ′ is obtained by replacing the matrix B with u times (u ≧ 1) and v times (v ≧ 1). The matrix A ^(u) is obtained by performing the same replacement on the matrix A as the u-time replacement of the rows for the matrix B. The matrix E ^(v) is obtained by performing the same replacement as the unit matrix E with v column replacements for the matrix B. The ordering unit 40 stores data representing the matrices A ^(u) and E ^(v) in the matrix storage unit 48 and stores data representing the matrix B ′ in the intermediate result storage unit 42 (step S702).

  Next, the LU decomposition unit 34 reads the data representing the matrix B ′ from the intermediate result storage unit 42, performs LU decomposition on the matrix B ′, and converts the data representing the lower triangular matrix L ′ and the upper triangular matrix U ′. The represented data is stored in the intermediate result storage unit 42 (step S703).

Next, the basic matrix decomposition unit 36 reads data representing the upper triangular matrix U ′ from the intermediate result storage unit 42, and converts the upper triangular matrix U ′ into the basic upper triangular matrices U ′ ₁ , U ′ ₂ ,. The data representing the upper triangular matrix U ′ ₁ , U ′ ₂ ,..., U ′ _m is stored in the intermediate result storage unit 42 by being decomposed into U ′ _m products. The basic matrix decomposition unit 36 reads data representing the lower triangular matrix L ′ from the intermediate result storage unit 42 and converts the lower triangular matrix L into the basic lower triangular matrices L ′ ₁ , L ′ ₂ ,. _The data representing the basic lower triangular matrix L ′ ₁ , L ′ ₂ ,..., L ′ _m is stored in the intermediate result storage unit 42 (step S704).

The inverse matrix calculator 43 stores the basic upper triangular matrices U ′ ₁ , U ′ ₂ ,..., U ′ _m and the basic lower triangular matrices L ′ ₁ , L ′ ₂ ,. 'reads the data representing the _m, the matrix _{U' 1, U '2,} ···, U' m, L '1, L' 2, ···, unpaired corner section elements of L _'m code by inverting the matrix _{U '1, U' 2,} ···, U 'm, L' 1, L '2, ···, L' inverse matrix U of _{m '1} ^-1, U' ₂ ⁻¹ ,..., U ′ _m ⁻¹ , L ′ ₁ ⁻¹ , L ′ ₂ ⁻¹ ,..., L ′ _m ⁻¹ are calculated and inverse matrices U ′ ₁ ⁻¹ , U ′ ₂ ^{− _{1, ···, U 'm -1}} , L' 1 -1, L '2 -1, ···, L' and stores the data representing the _m ^-1 in the matrix storage unit 48 (step S705).

(Encoded vector generation processing)
FIG. 13 is a flowchart of an encoded vector generation process according to Modification 1 of the second embodiment.

Referring to FIG. 13, the matrix product calculation control unit 124 determines that the order of the matrices in which data is stored in the matrix storage unit 48 is A ^(u) , L ′ ₁ ⁻¹ , L ′ ₂ ^−1,. .., the order between the matrices is set so that L ′ _m ⁻¹ , U ′ _m ⁻¹ , U ′ _m−1 ⁻¹ ,..., U ′ ₁ ⁻¹ , E ^(v) (Step S801).

Next, the matrix product calculation control unit 124 sets the control variable i to 1 (step S802).
The matrix product calculation control unit 124 instructs the matrix product calculation unit 41 to read out the data representing the i th (= 1) -th order matrix A ^(u) from the matrix storage unit 48, and the matrix A ^(u) And the product A ^(u) X of the input information vector X is calculated. The matrix product calculation unit 41 sets the vector that is the multiplication result as the intermediate result R, and stores the data representing the intermediate result R in the intermediate result storage unit 42 (step S803).

  Next, the matrix product calculation control unit 124 increments the control variable i (step S804).

When the control variable i is less than (2m + 2) (YES in step S805), the matrix product calculation control unit 124 instructs the matrix product calculation unit 41 to receive the i-th rank matrix (the matrix) from the matrix storage unit 48. causes read data representing the Mat _i), the intermediate result storage unit 42 intermediate results are stored in the by reading the data representing the R, matrices Mat _i and an intermediate result product Mat _i × R and R Let's calculate. The matrix product calculation unit 41 sets the vector that is the multiplication result as a new intermediate result R, and updates the data representing the intermediate result R in the intermediate result storage unit 42 (step S806). Thereafter, the processing from step S804 is repeated.

When the control variable i becomes (2m + 2) (NO in step S805), the matrix product calculation control unit 124 instructs the matrix product calculation unit 41 to receive the i-th rank matrix (Mat) from the matrix storage unit 48. _i ) and the data representing the intermediate result R stored in the intermediate result storage unit 42 are read, and the product Mat _i × R of the matrix Mat _i and the intermediate result R is calculated. . The matrix product calculation unit 41 outputs a vector Y ′ as a multiplication result to the sign inversion unit 99. The sign inversion unit 99 outputs the inverted vector Y ′ as the parity vector Y (step 807).

  Thereafter, the array unit 20 generates and outputs an n-bit encoded vector Z composed of a k-bit information vector X and an m-bit parity vector Y (step S808).

(Effects of Modification 1 of Second Embodiment)
As described above, according to the first modification of the second embodiment, the parity vector Y is calculated according to the equation (E23) without performing backward substitution at the time of the encoded vector generation process. The information vector X can be encoded. In particular, since matrix B is ordered, the inverse matrix U ′ _i ⁻¹ (i = ^{1 to} m) of the basic upper triangular matrix and the inverse matrix L ′ _i ⁻¹ (i = ^{1 to} m) of the basic lower triangular matrix are More than the inverse matrix U _i ⁻¹ (i = ^{1 to} m) of the basic upper triangular matrix and the inverse matrix L _i ⁻¹ (i = ^{1 to} m) of the basic lower triangular matrix of the first modification of the first embodiment Furthermore, the matrix has a smaller number of non-zero elements, and the amount of calculation of the equation (E23) can be further reduced compared to the equation (C9). Further, since the parity vector Y is calculated by sequentially executing the matrix multiplication on the right side of the equation (E23), the result of each matrix multiplication becomes a vector, and the capacity for storing the intermediate result can be reduced.

Further, at the time of matrix decomposition processing, without performing an operation with a large amount of calculation of calculating an inverse matrix of the matrix B, generation of matrices B ′, A ^(u) and E ^(v) by ordering of the matrix B, matrix B LU decomposition of ′, decomposition of LU decomposed upper triangular matrix into product of basic upper triangular matrix, decomposition of LU decomposed lower triangular matrix into product of basic lower triangular matrix, basic upper triangular matrix and basic lower triangular matrix Therefore, the matrix necessary for generating the parity vector Y can be calculated with a small amount of calculation.

[Third Embodiment]
The encoder of the third embodiment performs an operation modulo 2.

(Encoding algorithm)
The encoder receives an information vector X consisting of k information bits x1, x2,..., Xk and performs an operation modulo 2 so as to satisfy m A parity vector Y composed of bits y1, y2,.
H (X, Y) ^T = 0 (F1)
Here, H is a parity check matrix (hereinafter also referred to as a check matrix), which is a matrix of m rows and n columns (= k + m).
(X, Y) ^T represents transposition of a vector composed of an information vector and a parity vector.

Formula (F1) is modified as follows in the case of an operation modulo 2.
BY≡AX (F2)
Here, the matrix A is a matrix of m rows and k columns including the first to kth columns of the check matrix H. The matrix B is a matrix of m rows and m columns including the (k + 1) -th column to the n-th column of the parity check matrix H.

When the matrix B is subjected to LU decomposition, the formula (F2) is transformed into the following formula.
LUY≡AX (F3)
Here, L is a lower triangular matrix and U is an upper triangular matrix.

From the formula (F3), the column vector Y is expressed by the following formula.
Y≡U ⁻¹ L ⁻¹ AX (F4)

The upper triangular matrix U is decomposed into products of basic upper triangular matrices U _i (i = 1 to m) as in the following equation.
U≡U _m U _m-1 ... U ₂ U ₁ ... (F5)
From the equation (F5), the inverse matrix U ⁻¹ is represented by the following equation.

U ^-1 ≡ U ₁ ^-1 U ₂ ^-1 ... U _m-1 ^-1 U _m ^-1 ... (F6)

The lower triangular matrix L is decomposed into products of basic lower triangular matrices L _i (i = 1 to m) as shown in the following expression.
L≡L ₁ L ₂ ... L _m-1 L _m (F7)
From the equation (F7), the inverse matrix L ⁻¹ is represented by the following equation.
L ^-1 ≡L _m ^-1 L _m-1 ^-1 ... L ₂ ^-1 L ₁ ^-1 ... (F8)

Expression (F4) is transformed into the following expression by Expression (F6) and Expression (F8).
Y≡ (U ₁ ^-1 U ₂ ^-1 ... U _m-1 ^-1 U _m ^-1 ) (L _m ^-1 L _m-1 ^-1 ... L ₂ ^-1 L ₁ ^-1 ) AX .. (F9)

By the way, in the case of an operation modulo 2, the basic upper triangular matrix U _i (i = 1 to m) and the basic lower triangular matrix L _i (i = 1 to m) have the following properties.
U _i ≡ (U _i ) ^-1 (F10)
L _i ≡ (L _i ) ⁻¹ (F11)

Expression (F9) is transformed into the following expression by Expression (F10) and Expression (F11).
Y≡U ₁ U ₂ ... U _m-1 U _m L _m L _m-1 ... L ₂ L ₁ AX (F12)

Expression (F12) is a multiplication of (2 × m + 2) matrices, but if a certain condition is satisfied, it is assumed that the products of several matrices can be represented by one matrix. It is assumed that the product of t matrices of U _{1 to} U _t can be represented by one matrix C as in the following equation.
(U ₁ U ₂ ... U _t ) ≡C (F13)
In this case, the certain conditions, the matrix U _1, U _2, · · ·, or the number of non-zero elements of the non-diagonal terms of the non-zero elements sum matrix C of the number of off-diagonal terms of U _t Suppose that

From formula (F13), formula (F12) is expressed by the following formula.
Y≡CU _{t + 1} ... U _m-1 U _m L _m L _m-1 ... L ₂ L ₁ AX (F14)

Using the parity vector Y calculated by the equation (F14) and the information vector X, the encoded vector Z is expressed by the following equation.
Z≡ (X, Y) (F15)
Here, the first bit to the k-th bit of the encoded vector Z are bits of the information vector X, and the (k + 1) -th bit to the n-th bit of the encoded vector Z are bits of the parity vector Y.

(Constitution)
FIG. 14 is a diagram illustrating the configuration of the encoder according to the third embodiment.

  Referring to FIG. 14, the encoder 2e includes a parity check matrix decomposition unit 30, an LU decomposition unit 14, a basic matrix decomposition unit 16, and an arrangement unit 20, which are the same as those in the first embodiment. Description of these will not be repeated.

  The encoder 2e includes an intermediate result storage unit 52, a matrix storage unit 58, a matrix product calculation unit 51, a matrix product calculation control unit 125, and a matrix multiplication number reduction unit 55. In the following description, it is assumed that (t + 1) <m.

  The intermediate result storage unit 52 stores data representing an intermediate result in a process (matrix decomposition process) for generating a matrix included in the formula (F14). Further, the intermediate result storage unit 52 stores data representing the intermediate result of (2m−t + 2) times of matrix multiplication included in the equation (F14) during the generation process of the parity vector Y.

Matrix storage unit 58 represents the basic upper triangular matrix _{U t + 1, U t +} 2, ···, data representing the U _m, the fundamental lower triangular matrix _{L m, L m-1,} ···, a L ₁ Data and data representing the matrix A are stored.

The matrix product calculation unit 51 includes an adder 57 and a multiplier 59. In order to calculate the parity vector Y, the matrix product calculation unit 51 performs (2m−t + 2) matrix multiplications included in the formula (F14) in the order instructed by the matrix product calculation control unit 125. The matrix product calculation unit 51 receives the input information vector X, the matrices A, L ₁ , L ₂ ,..., L _m−1 , L _m , U _m , U _m−1,. Each matrix multiplication is executed using two of the data representing U _{t + 1} , C and the data representing the intermediate result in the intermediate result storage unit 52. The matrix product calculation unit 51 stores data representing the calculation result in the intermediate result storage unit 52 as data representing the intermediate result until (2m−t + 2) times of matrix multiplications included in the formula (F14) is completed. Let Further, the matrix product calculation unit 51 executes the matrix multiplication instructed by the matrix multiplication number reduction unit 55 when executing the process of reducing the number of matrix multiplications included in the formula (F12).

When the matrix multiplication number reduction unit 55 performs the process of reducing the number of matrix multiplications included in the formula (F12), the matrix order in which the data is stored in the intermediate result storage unit 52 is U ₁ , U ₂ ,..., U _m−1 , U _m , L _m , L _m−1 ,..., L ₂ , L ₁ , A are set in the order between the matrices, and in this order The matrix product calculation unit 51 is controlled to perform matrix multiplication. The matrix multiplication number reduction unit 55, in each multiplication, the number of non-zero elements in the off-diagonal terms of the two matrices to be multiplied and the number of non-zero elements in the off-diagonal terms of one matrix C that is the multiplication result When the difference between the latter and the former is equal to or smaller than a predetermined threshold a (a> 0), two matrices to be multiplied are replaced with one matrix C. When the matrix C is replaced in this way, the matrix C is multiplied by the next matrix.

On the other hand, when the difference between the latter and the former exceeds a predetermined threshold value a, the matrix multiplication number reduction unit 55 does not replace the two matrices to be multiplied with one matrix C, but instead replaces the matrices A, L ₁ , _{L 2, ···, L m-} 1, L m, U m, U m-1, ···, and the total number of non-zero elements of the non-diagonal terms of U _1, the matrix a, L _1, L ₂ ,..., L _m−1 , L _m , U _m , U _m−1 ,..., U ₁ excluding the replaced matrix and the non-diagonal terms of matrix C When the latter value is less than or equal to the former value, the matrix A, L ₁ , L ₂ ,..., L _m−1 , L _m , U _m , U _{m −1} ,..., U ₁ excluding the replaced matrix and the matrix C are stored in the matrix storage unit 58. If the matrix up to the t rank is substituted, the matrix _{C, U t + 1, ···} , U m-1, U m, L m, L m-1, ···, L 2, L ₁ and A are stored in the matrix storage unit 58.

Matrix product calculation control unit 125, in order to calculate the parity vector Y, A order from the head of the matrix data in the matrix storage unit 58 is _{stored, L 1, L 2, ···} , L m-1 , L _m , U _m , U _m−1 ,..., U _{t + 1} , C, the order between the matrices is set, and the matrix product calculation unit 51 performs matrix multiplication in this order. To control. Further, the matrix product calculation control unit 125 causes the matrix product calculation unit 51 to output the parity vector Y that is the calculation result after the (2m−t + 2) times of matrix multiplication included in the formula (F14).

(Matrix decomposition processing)
FIG. 15 is a flowchart of matrix decomposition processing according to the third embodiment.

Referring to FIG. 15, first, steps S101 to S103 of the first embodiment are performed by the check matrix decomposition unit 30, the matrix storage unit 58, the intermediate result storage unit 52, the LU decomposition unit 14, and the basic matrix decomposition unit 16. Executed. However, the generated matrices U ₁ , U ₂ ,..., U _m−1 , U _m , L _m , L _m−1 ,..., L ₂ , L ₁ , A are the same as those in the first embodiment. As described above, it is stored in the intermediate result storage unit 52, not in the matrix storage unit 58. (Step S901).

Next, the matrix multiplication number reduction unit 55 performs U ₁ , U ₂ ,..., U _m−1 , U _m , L _m , L on the matrix whose data is stored in the intermediate result storage unit 52 from the top. _The order is set so as to be in the order of _m−1 ,..., L ₂ , L ₁ , A (step S902).

Next, the matrix multiplication number reduction unit 55 sets the control variable t to 1 (step S903).
Next, matrix multiplication number reduction section 55, matrix storage unit 58 from the t (= 1) reads the data representing the matrix U ₁ rank, the matrix U ₁ as a matrix C, the data representing the matrix C intermediate results It memorize | stores in the memory | storage part 52 (step S904).

  The matrix number product reduction unit reads the data representing the matrix C from the intermediate result storage unit 52, counts the non-zero element number SC0 of the matrix C, and stores the non-zero element number SC0 in the intermediate result storage unit 52 (step). S905).

  Next, the matrix multiplication number reduction unit 55 reads data representing the (t + 1) -th rank matrix, and sets the matrix as the matrix mat (step S906).

  Next, the matrix multiplication number reduction unit 55 counts the non-zero element number SC1 of the non-diagonal term of the matrix mat, and the non-zero element number SC1 and the non-zero element number SC0 of the matrix C in the intermediate result storage unit 52 And the sum SC2 of the number of non-zero elements is stored in the intermediate result storage unit 52 (step S907).

  Next, the matrix product calculation unit 51 calculates the product C × Mat of the matrix C and the matrix Mat, and sets the calculation result as the matrix D (step S908).

  Next, the matrix multiplication number reduction unit 55 counts the non-zero element number SC3 of the non-diagonal term of the matrix D, and stores the non-zero element number SC3 in the intermediate result storage unit 52 (step S909).

  The matrix multiplication number reduction unit 55 reads the non-zero element number SC2 and the non-zero element number SC3 from the intermediate result storage unit 52, and determines whether the difference between SC3 and SC2 is equal to or less than the threshold value a (step). S910).

  If the difference between SC3 and SC2 is equal to or less than the threshold value a (YES in step S910), the matrix multiplication number reduction unit 55 increments the control variable t (step S911), and the matrix C in the intermediate result storage unit 52 Is updated with the matrix D (step S912), the number of non-zero elements SC0 in the intermediate result storage unit 52 is updated with the number of non-zero elements SC3, and the processing from step S906 is repeated.

On the other hand, when the difference between SC3 and SC2 exceeds the threshold value a (NO in step S910) and the control variable t is 2 or more (YES in step S914), the matrix multiplication number reduction unit 55 determines the matrix storage unit 58. , U _m−1 , U _m , L _m , L _m−1 ,..., L ₂ , L ₁ , A are unpaired with respect to each of the matrices U ₁ , U ₂ ,. The number of non-zero elements in the angular term is counted, and the sum SC4 of these non-zero elements is stored in the intermediate result storage unit 52 (step S915).

Next, the matrix multiplication number reduction unit 55 is configured to store the matrixes U ₁ , U ₂ ,..., U _m−1 , U _m , L _m , L _m−1 , ..., L ₂ , L ₁ , A. The number of non-zero elements in the non-diagonal terms is counted for each of the matrices after the (t + 1) th rank, and the total number SC5 of the non-zero elements and the number of non-zero elements of the matrix C in the intermediate result storage unit 52 are counted. The sum SC6 with SC0 is calculated, and the sum SC6 of the number of non-zero elements is stored in the intermediate result storage unit 52 (step S916).

  The matrix multiplication number reduction unit 55 reads the sum SC4 of the number of nonzero elements and the sum SC6 of the number of nonzero elements from the intermediate result storage unit 52, and determines whether SC6 is SC4 or less (step S917).

When the SC6 is SC4 or less (YES in step S917), the matrix multiplication number reduction unit 55 determines the matrices U ₁ , U ₂ ,..., U _m−1 , U _m , in the intermediate result storage unit 52. L _m , L _m−1 ,..., L ₂ , L ₁ , A matrix U _{t + 1} , U _{t + 2} ,. U _m−1 , U _m , L _m , L _m−1 ,..., L ₂ , L ₁ , A and the matrix C are stored in the matrix storage unit 58 (step S918).

(Encoded vector generation processing)
FIG. 16 is a flowchart of encoded vector generation processing according to the third embodiment.

Referring to FIG. 16, matrix product calculation control section 125 determines that the order of the matrices in which data is stored in matrix storage section 58 is A, L ₁ , L ₂ ,..., L _m−1 , L The order between the matrices is set so that _m , U _m , U _m−1 ,..., U _{t + 1} , C are in order (step S1001).

  Next, the matrix product calculation control unit 125 sets the control variable i to 1 (step S1002).

  The matrix product calculation control unit 125 instructs the matrix product calculation unit 51 to read out the data representing the i-th (= 1) -th rank matrix A from the matrix storage unit 58, and the matrix A and the input information The product AX with the vector X is calculated. The matrix product calculation unit 51 sets the vector that is the multiplication result as the intermediate result R, and stores the data representing the intermediate result R in the intermediate result storage unit 52 (step S1003).

  Next, the matrix product calculation control unit 125 increments the control variable i (step S1004).

If the control variable i is less than (2m−t + 2) (YES in step S1005), the matrix product calculation control unit 125 instructs the matrix product calculation unit 51 to receive the i-th rank matrix (from the matrix storage unit 58). causes read data representing this and Mat _i), by reading the data representing the intermediate results R stored in the intermediate result storage unit 52, the product Mat _i the matrix Mat _i and an intermediate result R XR is calculated. The matrix product calculation unit 51 sets the vector that is the multiplication result as a new intermediate result R, and updates the data representing the intermediate result R in the intermediate result storage unit 52 (step S1006). Thereafter, the processing from step S1004 is repeated.

When the control variable i becomes (2m−t + 2) (NO in step S1005), the matrix product calculation control unit 125 instructs the matrix product calculation unit 51 to receive the i-th rank matrix from the matrix storage unit 58. Data representing (Mat _i ) is read out, and data representing the intermediate result R stored in the intermediate result storage unit 52 is read out to obtain a product Mat _i × R of the matrix Mat _i and the intermediate result R. Let it be calculated. The matrix product calculation unit 51 outputs the vector that is the multiplication result as the parity vector Y (step S1007).

  Thereafter, the array unit 20 generates and outputs an n-bit encoded vector Z composed of a k-bit information vector X and an m-bit parity vector Y (step S1008).

(Effect of the third embodiment)
As described above, according to the third embodiment, since the parity vector Y is calculated according to the equation (F14) without performing backward substitution in the encoded vector generation process, the information vector X can be calculated with a small amount of calculation. Encoding can be performed. In particular, since the basic upper triangular matrix U _i (i = t + 1 to m) and the basic lower triangular matrix L _i (i = 1 to m) are matrices with a small number of non-zero elements, the amount of calculation of Expression (F14) is Few. In the equation (F14), U _{1 to} U _{t in} the equation (B12) of the first embodiment are represented by one matrix C, and the number of non-zero elements in the off-diagonal terms of the matrix C is the matrix U ₁ to Since it is less than or equal to the sum of the number of non-zero elements of the off-diagonal term of U _t , the calculation of the formula (F14) has a smaller calculation amount than the calculation of the formula (B12) of the first embodiment. Further, since the parity vector Y is calculated by sequentially executing from the matrix multiplication on the right side of the equation (F14), the result of each matrix multiplication becomes a vector, and the capacity for storing the intermediate result can be reduced.

  Further, at the time of matrix decomposition processing, without performing an operation with a large amount of calculation of calculating an inverse matrix of the matrix B, LU decomposition of the matrix B, decomposition of the upper triangular matrix subjected to LU decomposition into a product of the basic upper triangular matrix Since it is sufficient to perform the decomposition of the LU-decomposed lower triangular matrix into the product of the basic lower triangular matrix and the reduction processing of the matrix multiplication number, the matrix necessary for generating the parity vector Y with a small amount of calculation is calculated. Can do.

[Modification 1 of the third embodiment]
The encoder of the first modification of the third embodiment performs a real number operation.

(Encoding algorithm)
The encoder receives an information vector X composed of k information bits x1, x2,..., Xk and performs a real number operation to thereby obtain m parity bits y1, y2 satisfying the following expression. ,..., Ym to generate a parity vector Y.
H (X, Y) ^T = 0 (G1)
Here, H is a parity check matrix (hereinafter also referred to as a check matrix), which is a matrix of m rows and n columns (= k + m).
(X, Y) ^T represents transposition of a vector composed of an information vector and a parity vector.

Expression (G1) is modified as follows in the case of real number calculation.
BY = −AX (G2)
Here, the matrix A is a matrix of m rows and k columns including the first to kth columns of the check matrix H. The matrix B is a matrix of m rows and m columns including the (k + 1) -th column to the n-th column of the parity check matrix H.

When the matrix B is subjected to LU decomposition, the equation (G2) is transformed into the following equation.
LUY = −AX (G3)
Here, L is a lower triangular matrix and U is an upper triangular matrix.
From the equation (G3), the column vector Y is expressed by the following equation.
Y = −U ⁻¹ L ⁻¹ AX (G4)

The upper triangular matrix U is decomposed into products of basic upper triangular matrices U _i (i = 1 to m) as in the following equation.
U = U _m U _m-1 ... U ₂ U ₁ ... (G5)
From Formula (G5), U ⁻¹ is represented by the following formula.
U ⁻¹ = U ₁ ⁻¹ U ₂ ⁻¹ ... U _m-1 ⁻¹ U _m ⁻¹ ... (G6)
Here, U _i ⁻¹ is obtained by inverting the sign of the element of U _{i in which} the element of the diagonal term is 1 and the element of the off-diagonal term.

The lower triangular matrix L is decomposed into products of basic lower triangular matrices L _i (i = 1 to m) as shown in the following expression.
L = L ₁ L ₂ ... L _m-1 L _m (G7)
From the formula (G7), L ⁻¹ is represented by the following formula.
L ⁻¹ = L _m ⁻¹ L _m−1 ⁻¹ ... L ₂ ⁻¹ L ₁ ⁻¹ ... (G8)
Here, L _i ⁻¹ is obtained by inverting the sign of the element of L _{i in which} the element of the diagonal term is 1 and the element of the non-diagonal term.

Expression (G4) is transformed into the following expression by Expression (G6) and Expression (G8).
Y = −U ₁ ⁻¹ U ₂ ⁻¹ ... U _m-1 ⁻¹ U _m ⁻¹ L _m ⁻¹ L _m−1 ⁻¹ ... L ₂ ⁻¹ L ₁ ⁻¹ AX. G9)
Equation (G9) is a multiplication of (2 × m + 2) matrices, but if certain conditions are satisfied, the product of several matrices can be represented by one matrix. It is assumed that the product of t matrices from U ₁ ^{−1 to} U _t ⁻¹ can be represented by one matrix C as in the following equation.
(U ₁ ⁻¹ U ₂ ⁻¹ ... U _t ⁻¹ ) ≡C (G10)
In this case, the certain conditions, the matrix ^{_{U 1 -1, U 2 -1,}} ···, the sum of the number of non-zero elements of the non-diagonal terms of U _t ^-1 is non-diagonal terms of the matrix C It is assumed that the number is greater than or equal to the number of non-zero elements.

From the equation (G10), the equation (G9) is expressed by the following equation.
Y = −CU _{t + 1} ⁻¹ ... U _m-1 ⁻¹ U _m ⁻¹ L _m ⁻¹ L _m−1 ⁻¹ ... L ₂ ⁻¹ L ₁ ⁻¹ AX (G11)

Using the parity vector Y calculated by the equation (G11) and the information vector X, the encoded vector Z is expressed by the following equation.
Z = (X, Y) (G12)
Here, the first bit to the k-th bit of the encoded vector Z are bits of the information vector X, and the (k + 1) -th bit to the n-th bit of the encoded vector Z are bits of the parity vector Y.

(Constitution)
FIG. 17 is a diagram illustrating a configuration of an encoder according to Modification 1 of the third embodiment.

  Referring to FIG. 17, this encoder 2f includes a parity check matrix decomposition unit 30, an LU decomposition unit 14, a basic matrix decomposition unit 16, and an inverse matrix calculation similar to those of the first modification of the first embodiment. The unit 23 and the arrangement unit 20 are provided. Description of these will not be repeated.

  The encoder 2 f includes an intermediate result storage unit 62, a matrix storage unit 68, a matrix product calculation unit 61, a matrix product calculation control unit 126, and a matrix multiplication number reduction unit 65. In the following description, it is assumed that (t + 1) <m.

  The intermediate result storage unit 62 stores data representing an intermediate result in a process (matrix decomposition process) for generating a matrix included in the equation (G11). Further, the intermediate result storage unit 62 stores data representing the intermediate result of (2m−t + 2) times of matrix multiplication included in the equation (G11) during the generation process of the parity vector Y.

The matrix storage unit 68 includes data representing the inverse matrices U _{t + 1} ⁻¹ , U _{t + 2} ⁻¹ ,..., U _m ⁻¹ of the basic upper triangular matrix, and the inverse matrix L _m ⁻¹ of the basic lower triangular matrix. , L _m−1 ⁻¹ ,..., L ₁ ⁻¹ and data representing the matrix A are stored.

The matrix product calculation unit 61 includes an adder 67 and a multiplier 69. In order to calculate the parity vector Y, the matrix product calculation unit 61 performs (2m−t + 2) times of matrix multiplications included in the equation (G11) in the order instructed by the matrix product calculation control unit 126. The matrix product calculation unit 61 receives the input information vector X, the matrices A, L ₁ ⁻¹ , L ₂ ⁻¹ ,..., L _m−1 ⁻¹ , L _m ⁻¹ , U in the matrix storage unit 68. Each matrix using two of _m ⁻¹ , U _m−1 ⁻¹ ,..., U _{t + 1} ⁻¹ , data representing C, and data representing an intermediate result in the intermediate result storage unit 62 Perform multiplication. The matrix product calculation unit 61 stores data representing the calculation result in the intermediate result storage unit 62 as data representing the intermediate result until (2m−t + 2) times of matrix multiplications included in the equation (G11) is completed. Let Further, the matrix product calculation unit 61 executes matrix multiplication instructed by the matrix multiplication number reduction unit 65 when executing the process of reducing the number of matrix multiplications included in the equation (G9).

When the matrix multiplication number reduction unit 65 performs the process of reducing the number of matrix multiplications included in the equation (G9), the order of the matrices in which the data is stored in the intermediate result storage unit 62 is U ₁ ⁻¹ from the top. , U ₂ ⁻¹ ,..., U _m-1 ⁻¹ , U _m ⁻¹ , L _m ⁻¹ , L _m−1 ⁻¹ ,..., L ₂ ⁻¹ , L ₁ ⁻¹ , A The order between the matrices is set so as to be in order, and the matrix product calculation unit 61 is controlled to perform matrix multiplication in this order. In each multiplication, the matrix multiplication number reduction unit 65 calculates the number of non-zero elements in the off-diagonal terms of the two matrices to be multiplied and the number of non-zero elements in the off-diagonal terms of one matrix C that is the multiplication result. When the difference between the latter and the former is equal to or smaller than a predetermined threshold a (a> 0), two matrices to be multiplied are replaced with one matrix C. When the matrix C is replaced in this way, the matrix C is multiplied by the next matrix.

On the other hand, when the difference between the latter and the former exceeds a predetermined threshold value a, the matrix multiplication number reduction unit 65 does not replace the two matrices to be multiplied with one matrix C, but instead performs the matrix A, L ₁ ^{− _{1, L 2 -1, ···,}} L m-1 -1, L m -1, U m -1, U m-1 -1, ···, of the off-diagonal terms of U ₁ ^-1 non Total number of zero elements and matrices A, L ₁ ⁻¹ , L ₂ ⁻¹ ,..., L _m−1 ⁻¹ , L _m ⁻¹ , U _m ⁻¹ , U _m−1 ⁻¹ ,.・ When the matrix of U ₁ ^-1 excluding the replaced matrix is compared with the sum of the number of non-zero elements in the off-diagonal terms of matrix C, and the latter value is less than or equal to the former value the matrix ^{_{A, L 1 -1, L 2}} -1, ···, L m-1 -1, L m -1, U m -1, U m-1 -1, ···, U 1 - ^The matrix storage unit 68 stores the matrix excluding the replaced matrix of 1 and the matrix C. When the matrix up to the t-th rank is replaced, the matrix C, U _{t + 1} ⁻¹ ,..., U _m−1 ⁻¹ , U _m ⁻¹ , L _m ⁻¹ , L _m−1 ^{− 1} ,..., L ₂ ⁻¹ , L ₁ ⁻¹ , A are stored in the matrix storage unit 68.

In order to calculate the parity vector Y, the matrix product calculation control unit 126 determines that the order of the matrices in which data is stored in the matrix storage unit 68 is A, L ₁ ⁻¹ , L ₂ ⁻¹ ,. The order between the matrices is set so that L _m-1 ⁻¹ , L _m ⁻¹ , U _m ⁻¹ , U _m−1 ⁻¹ ,..., U _{t + 1} ⁻¹ , C are in this order. The matrix product calculation unit 61 is controlled to perform matrix multiplication in this order. Further, the matrix product calculation control unit 126, after completing (2m−t + 2) times of matrix multiplication included in the equation (G11), outputs a vector Y ′ (= CU _{t + 1} ^{−) as} a calculation result from the matrix product calculation unit 61. ¹ ... U _m−1 ⁻¹ U _m ⁻¹ L _m ⁻¹ L _m−1 ⁻¹ ... L ₂ ⁻¹ L ₁ ⁻¹ AX) is output to the sign inversion unit 99.

(Matrix decomposition processing)
FIG. 18 is a flowchart of matrix decomposition processing according to Modification 1 of the third embodiment.

Referring to FIG. 18, first, Steps S201 to S204 of Modification 1 of the first embodiment are the check matrix decomposition unit 30, the matrix storage unit 68, the intermediate result storage unit 62, the LU decomposition unit 14, the basic matrix decomposition. This is executed by the unit 16 and the inverse matrix calculation unit 23. However, the generated matrices _{^{U 1 -1, U 2 -1,}} ···, U m-1 -1, U m -1, L m -1, L m-1 -1, ···, L 2 ⁻¹ , L ₁ ⁻¹ , and A are stored not in the matrix storage unit 68 but in the intermediate result storage unit 62 as in the first modification of the first embodiment. (Step S1101).

Next, the matrix multiplication number reduction unit 65 performs U ₁ ⁻¹ , U ₂ ⁻¹ ,..., U _m−1 ⁻¹ , U _m on the matrix whose data is stored in the intermediate result storage unit 62. _The order is set so as to be in the order of _m ⁻¹ , L _m ⁻¹ , L _m−1 ⁻¹ ,..., L ₂ ⁻¹ , L ₁ ⁻¹ , A (step S1102).

Next, the matrix multiplication number reduction unit 65 sets the control variable t to 1 (step S1103).
Next, the matrix multiplication number reduction unit 65 reads data representing the matrix U ₁ ⁻¹ of the t (= 1) th rank from the matrix storage unit 68 and represents the matrix C with the matrix U ₁ ⁻¹ as the matrix C. Data is stored in the intermediate result storage unit 62 (step S1104).

  The matrix number product reduction unit reads the data representing the matrix C from the intermediate result storage unit 62, counts the non-zero element number SC0 of the matrix C, and stores the non-zero element number SC0 in the intermediate result storage unit 62 (step). S1105).

  Next, the matrix multiplication number reduction unit 65 reads data representing the (t + 1) -th rank matrix and sets the matrix as the matrix mat (step S1106).

  Next, the matrix multiplication number reduction unit 65 counts the non-zero element number SC1 of the non-diagonal term of the matrix mat, and the non-zero element number SC1 and the non-zero element number SC0 of the matrix C in the intermediate result storage unit 62 And the sum SC2 of the number of non-zero elements is stored in the intermediate result storage unit 62 (step S1107).

  Next, the matrix product calculation unit 61 calculates the product C × Mat of the matrix C and the matrix Mat, and sets the calculation result as the matrix D (step S1108).

  Next, the matrix multiplication number reduction unit 65 counts the non-zero element number SC3 of the non-diagonal term of the matrix D, and stores the non-zero element number SC3 in the intermediate result storage unit 62 (step S1109).

  The matrix multiplication number reduction unit 65 reads the sum SC2 of the number of non-zero elements and the number of non-zero elements SC3 from the intermediate result storage unit 62, and determines whether the difference between SC3 and SC2 is less than or equal to the threshold value a (step). S1110).

  If the difference between SC3 and SC2 is equal to or smaller than the threshold value a (YES in step S1110), the matrix multiplication number reduction unit 65 increments the control variable t (step S1111), and the matrix C in the intermediate result storage unit 62 Is updated with the matrix D (step S1112), the number of nonzero elements SC0 in the intermediate result storage unit 62 is updated with the number of nonzero elements SC3, and the processing from step S1106 is repeated.

On the other hand, when the difference between SC3 and SC2 exceeds the threshold value a (NO in step S1110) and the control variable t is 2 or more (YES in step S1114), the matrix multiplication number reduction unit 65 determines the matrix storage unit 68. matrix U ₁ ^-1 where data is stored ^{_{in, U 2 -1, ···, U}} m-1 -1, U m -1, L m -1, L m-1 -1, ···, The number of non-zero elements of non-diagonal terms is counted for each of L ₂ ⁻¹ , L ₁ ⁻¹ , and A, and the sum SC4 of these non-zero elements is stored in the intermediate result storage unit 62 (step S1115).

Next, the matrix multiplication number reduction unit 65 includes matrices U ₁ ⁻¹ , U ₂ ⁻¹ ,..., U _m−1 ⁻¹ , U _m ⁻¹ , L _m ⁻¹ , L _m−1 ⁻¹ , ..., the number of non-zero elements of non-diagonal terms is counted for each of the matrices after the (t + 1) -th rank among L ₂ ^-1 , L ₁ ^-1 and A, and the sum of the numbers of these non-zero elements The sum SC6 of SC5 and the number of nonzero elements SC0 of the matrix C in the intermediate result storage unit 62 is calculated, and the sum SC6 of the number of nonzero elements is stored in the intermediate result storage unit 62 (step S1116).

  The matrix multiplication number reduction unit 65 reads out the sum SC4 of the number of non-zero elements and the sum SC6 of the number of non-zero elements from the intermediate result storage unit 62, and determines whether SC6 is SC4 or less (step S1117).

When SC6 is equal to or lower than SC4 (YES in step S1117), the matrix multiplication number reduction unit 65 determines the matrices U ₁ ⁻¹ , U ₂ ⁻¹ ,..., U _{m−1 in the} intermediate result storage unit 62. ^-1 , U _m ⁻¹ , L _m ⁻¹ , L _m−1 ⁻¹ ,..., L ₂ ⁻¹ , L ₁ ⁻¹ , excluding the matrix from the first rank to the t rank matrix ^{_{U t + 1 -1, U t}} + 2 -1, ···, U m-1 -1, U m -1, L m -1, L m-1 -1, ···, L 2 ⁻¹ , L ₁ ⁻¹ , A, and the matrix C are stored in the matrix storage unit 68 (step S1118).

(Encoded vector generation processing)
FIG. 19 is a flowchart of an encoded vector generation process according to Modification 1 of the third embodiment.

Referring to FIG. 19, the matrix product calculation control unit 126 determines that the order of the matrices in which data is stored in the matrix storage unit 68 is A, L ₁ ⁻¹ , L ₂ ⁻¹ _,. The order between the matrices is set so that ₋₁ ⁻¹ , L _m ⁻¹ , U _m ⁻¹ , U _m−1 ⁻¹ ,..., U _{t + 1} ⁻¹ , C are in this order (step S1201). ).

  Next, the matrix product calculation control unit 126 sets the control variable i to 1 (step S1202).

  The matrix product calculation control unit 126 instructs the matrix product calculation unit 61 to read out the data representing the i-th (= 1) -th rank matrix A from the matrix storage unit 68, and the matrix A and the input information The product AX with the vector X is calculated. The matrix product calculation unit 61 sets the vector that is the multiplication result as the intermediate result R, and stores the data representing the intermediate result R in the intermediate result storage unit 62 (step S1203).

  Next, the matrix product calculation control unit 126 increments the control variable i (step S1204).

If the control variable i is less than (2m−t + 2) (YES in step S1205), the matrix product calculation control unit 126 instructs the matrix product calculation unit 61 to transmit the i-th rank matrix from the matrix storage unit 68 ( causes read data representing this and Mat _i), by reading the data representing the intermediate results R stored in the intermediate result storage unit 62, the product Mat _i the matrix Mat _i and an intermediate result R XR is calculated. The matrix product calculation unit 61 sets the vector that is the multiplication result as a new intermediate result R, and updates the data representing the intermediate result R in the intermediate result storage unit 62 (step S1206). Thereafter, the processing from step S1204 is repeated.

When the control variable i becomes (2m−t + 2) (NO in step S1205), the matrix product calculation control unit 126 instructs the matrix product calculation unit 61 to receive the i-th rank matrix from the matrix storage unit 68. Data representing (Mat _i ) is read out, and data representing the intermediate result R stored in the intermediate result storage unit 62 is read out to obtain a product Mat _i × R of the matrix Mat _i and the intermediate result R. Let it be calculated. The matrix product calculation unit 61 outputs a vector Y ′ as a multiplication result to the sign inversion unit 99. The sign inversion unit 99 outputs the inverted vector Y ′ as the parity vector Y (step S1207).

  After that, the array unit 20 generates and outputs an n-bit encoded vector Z composed of a k-bit information vector X and an m-bit parity vector Y (step S1208).

(Effect of Modification 1 of Third Embodiment)
As described above, according to the first modification of the third embodiment, the parity vector Y is calculated according to the equation (G11) without performing backward substitution at the time of encoded vector generation processing. The information vector X can be encoded. In particular, the inverse matrix U _i ⁻¹ (i = t + ^{1 to} m) of the basic upper triangular matrix and the inverse matrix L _i ⁻¹ (i = ^{1 to} m) of the basic lower triangular matrix are matrices having a small number of nonzero elements. Therefore, the calculation amount of the formula (G11) is small. In Expression (G11), U ₁ ^{−1 to} U _t ⁻¹ of Expression (C9) of Modification 1 of the first embodiment are represented by one matrix C, and the non-diagonal terms of the matrix C are non-diagonal. Since the number of zero elements is equal to or less than the sum of the number of non-zero elements in the non-diagonal terms of the matrices U ₁ ^{−1 to} U _t ⁻¹ , the calculation of Expression (G11) The calculation amount is smaller than the calculation of (C9). Further, since the parity vector Y is calculated by sequentially executing the matrix multiplication on the right side of the equation (G11), the result of each matrix multiplication becomes a vector, and the capacity for storing the intermediate result can be reduced.

  Further, at the time of matrix decomposition processing, without performing an operation with a large amount of calculation of calculating an inverse matrix of the matrix B, LU decomposition of the matrix B, decomposition of the upper triangular matrix subjected to LU decomposition into a product of the basic upper triangular matrix Since the decomposition of the LU-decomposed lower triangular matrix into the product of the basic lower triangular matrix, the inversion of the signs of the non-diagonal terms of the basic upper triangular matrix and the basic lower triangular matrix, and the matrix multiplication number reduction processing may be performed. A matrix necessary for generating the parity vector Y can be calculated with a small amount of calculation.

[Fourth Embodiment]
The encoder of the fourth embodiment performs an operation modulo 2.

(Encoding algorithm)
The encoder receives an information vector X consisting of k information bits x1, x2,..., Xk and performs an operation modulo 2 so as to satisfy m A parity vector Y composed of bits y1, y2,.
H (X, Y) ^T ≡0 (H1)
Here, H is a parity check matrix (hereinafter also referred to as a check matrix), which is a matrix of m rows and n columns (= k + m).
(X, Y) ^T represents transposition of a vector composed of an information vector and a parity vector.

Expression (H1) is transformed as follows in the case of an operation modulo 2.
BY≡AX (H2)
Here, the matrix A is a matrix of m rows and k columns including the first to kth columns of the check matrix H. The matrix B is a matrix of m rows and m columns including the (k + 1) -th column to the n-th column of the parity check matrix H.

Assume that ordering is performed on the matrix B as follows.
First, when two rows of the matrix B are exchanged, the expression (H2) is transformed as follows.
P ₁ BY≡P ₁ AX (H3)
Here, P ₁ is a transformation matrix representing the first row replacement. In the case of converting i row and j row, P ₁ is a matrix in which the i row and j row of the unit matrix E are exchanged.

Further, when the two columns of the matrix B are exchanged, the expression (H3) is transformed as follows.
P ₁ BQ ₁ Q ₁ ^-1 Y≡P ₁ AX (H4)
Here, Q ₁ is a transformation matrix representing the first column replacement. When converting the i column and the j column, Q ₁ is a matrix in which the i column and the j column of the unit matrix E are exchanged.

Define a new matrix as follows:
B ⁽¹⁾ ≡P ₁ BQ ₁ (H5)
A ⁽¹⁾ ≡ P ₁ A (H6)
From the formula (H5) and the formula (H6), the formula (H4) is represented by the following formula.
B ⁽¹⁾ Q ₁ ^-1 Y≡A ⁽¹⁾ X ⁾ ... (H7)

Further, when two rows of the matrix B ⁽¹⁾ are exchanged, the expression (H7) is transformed as follows.
_{^{P 2 B (1) Q 1}} -1 Y≡P 2 A (1) X ··· (H8)
Here, P ₂ is a transformation matrix representing the second row replacement.

Further, when the two columns of the matrix B ⁽¹⁾ are exchanged, the expression (H8) is transformed as follows.
P ₂ B ⁽¹⁾ Q ₂ Q ₂ ^-1 Q ₁ ^-1 Y≡P ₂ A ⁽¹⁾ X (H9)
Here, Q ₂ is a transformation matrix representing the second column replacement.

Define a new matrix as follows:
B ⁽²⁾ ≡ P ₂ B ⁽¹⁾ Q ₂ (H10)
A ⁽²⁾ ≡ P ₂ A ⁽¹⁾ (H11)
From the equations (H10) and (H11), the equation (H9) is represented by the following equation.
B ⁽²⁾ Q ₂ ^-1 Q ₁ ^-1 Y≡A ⁽²⁾ X ... (H12)

Similarly, for the matrix B, the row replacement and the v-th (v ≧ 1) extra column are performed for the u-th (u ≧ 1). Assuming that the matrix representing the s-th (1 ≦ s ≦ u) row replacement is P _s and the matrix representing the s-th (1 ≦ s ≦ v) column replacement is Q _s , the following new matrix Define
B′≡P _u ... P ₂ P ₁ BQ ₁ Q ₂ ... Q _v (H13)
A ^(u) ≡P _u ... P ₂ P ₁ A (H14)
E ^(v) ≡Q ₁ Q ₂ ... Q _v ... (H15)
The following expressions are established by Expression (H13), Expression (H14), and Expression (H15).
B′E ^{(v) -1} Y≡A ^(u) X (H16)

When the matrix B ′ is subjected to LU decomposition, Expression (H16) is transformed into the following expression.
L′ U′E ^{(v) -1} Y≡A ^(u) X (H17)
Here, L ′ is a lower triangular matrix and U ′ is an upper triangular matrix.

From the equation (H17), the column vector Y is expressed by the following equation.
Y≡E ^(v) U ′ ⁻¹ L ′ ⁻¹ A ^(u)) X (H18)

The upper triangular matrix U ′ is decomposed into products of basic upper triangular matrices U ′ _i (i = 1 to m) as in the following equation.
_{U'≡U 'm U' m-1} ··· U '2 U' 1 ··· (H19)
From the formula (H19), U ′ ⁻¹ is represented by the following formula.

U ' ^-1 ≡ U' ₁ ^-1 U ' ₂ ^-1 ... U' _m-1 ^-1 U ' _m ^-1 ... (H20)

The lower triangular matrix L ′ is decomposed into products of basic lower triangular matrices L ′ _i (i = 1 to m) as in the following equation.
L′ ≡L ′ ₁ L ′ ₂ ... L ′ _m−1 L ′ _m (H21)
From the formula (H21), L ′ ⁻¹ is represented by the following formula.
L ′ ⁻¹ ≡L ′ _m ⁻¹ L ′ _m−1 ⁻¹ ... L ′ ₂ ⁻¹ L ′ ₁ ⁻¹ ... (H22)

Expression (H18) is transformed into the following expression by Expression (H20) and Expression (H22).
Y≡E ^(v) (U ′ ₁ ⁻¹ U ′ ₂ ⁻¹ ... U ′ _m-1 ⁻¹ U ′ _m ⁻¹ ) (L ′ _m ⁻¹ L ′ _m−1 ⁻¹ ... L ' ₂ ^-1 L' ₁ ^-1 ) A ^(u) X (H23)

By the way, in the case of operation modulo 2, the basic upper triangular matrix U ′ _i (i = 1 to m) and the basic lower triangular matrix L ′ _i (i = 1 to m) have the following properties. Have.
U ′ _i ≡ (U ′ _i ) ⁻¹ (H24)
L ′ _i ≡ (L ′ _i ) ⁻¹ (H25)
Expression (H23) is transformed into the following expression by Expression (H24) and Expression (H25).
Y≡E ^(v) U ′ ₁ U ′ ₂ ... U ′ _m−1 U ′ _m L ′ _m L ′ _m−1 ... L ′ ₂ L ′ ₁ A ^(u) X (H26 )

Expression (H26) is a multiplication of (2 × m + 3) matrices, but if certain conditions are satisfied, it is assumed that the products of several matrices can be represented by one matrix. It is assumed that the product of t matrices of E ^(v) and U ′ _{1 to} U ′ _t−1 can be represented by one matrix C as in the following equation.
E ^(v) (U ′ ₁ U ′ ₂ ... U ′ _t-1 ) ≡C (H27)
In this case, the certain conditions, the matrix _{^{E (v), U '1}} , U' 2, ···, U 't-1 of the non-diagonal terms of the non-zero elements of the number non sum of matrix C of It is assumed that the number is greater than or equal to the number of nonzero elements in the diagonal term.

From formula (H27), formula (H26) is expressed by the following formula.
Y≡CU ' _t ... U' _m-1 U ' _m L' _m L ' _m-1 ... L' ₂ L ' ₁ A ^(u) X (H28)

Using the parity vector Y calculated by the equation (H28) and the information vector X, the encoded vector Z is expressed by the following equation.
Z≡ (X, Y) (H29)
Here, the first bit to the k-th bit of the encoded vector Z are bits of the information vector X, and the (k + 1) -th bit to the n-th bit of the encoded vector Z are bits of the parity vector Y.

(Constitution)
FIG. 20 is a diagram illustrating a configuration of an encoder according to the fourth embodiment.

  Referring to FIG. 20, this encoder 2g includes a parity check matrix decomposing unit 30, an ordering unit 40, an LU decomposing unit 34, a basic matrix decomposing unit 36, and an array unit similar to those in the second embodiment. 20. Description of these will not be repeated.

  The encoder 2g includes an intermediate result storage unit 72, a matrix storage unit 78, a matrix product calculation unit 71, a matrix product calculation control unit 127, and a matrix multiplication number reduction unit 75. In the following description, it is assumed that t <m.

  The intermediate result storage unit 72 stores data representing an intermediate result in a process (matrix decomposition process) for generating a matrix included in the formula (H28). Further, the intermediate result storage unit 72 stores data representing the intermediate result of (2m−t + 3) times of matrix multiplication included in the equation (H28) during the generation process of the parity vector Y.

The matrix storage unit 78 stores data representing basic upper triangular matrices U ′ _t , U ′ _{t + 1} ,..., U ′ _m , basic lower triangular matrices L ′ _m , L ′ _m−1 ,. Data representing ' ₁ and data representing matrices C and A ^(u) are stored.

The matrix product calculation unit 71 includes an adder 77 and a multiplier 79. In order to calculate the parity vector Y, the matrix product calculation unit 71 performs (2m−t + 3) times of matrix multiplications included in the formula (H28) in the order instructed by the matrix product calculation control unit 127. The matrix product calculation unit 71 receives the input information vector X, the matrix A ^(u) , L ′ ₁ , L ′ ₂ ,..., L ′ _m−1 , L ′ _m , U ′ in the matrix storage unit 78. Each matrix multiplication is executed using two of data representing _m , U ′ _m−1 ,..., U ′ _t , and C, and data representing an intermediate result in the intermediate result storage unit 72. The matrix product calculation unit 71 stores data representing the calculation result in the intermediate result storage unit 72 as data representing the intermediate result until (2m−t + 3) times of matrix multiplications included in the formula (H28) is completed. Let In addition, the matrix product calculation unit 71 executes matrix multiplication instructed by the matrix multiplication number reduction unit 75 when executing the process of reducing the number of matrix multiplications included in the formula (H26).

When the matrix multiplication number reduction unit 75 performs the process of reducing the number of matrix multiplications included in the formula (H26), the order of the matrices in which the data is stored in the intermediate result storage unit 72 is E ^(v) , U ′ ₁ , U ′ ₂ ,..., U ′ _m−1 , U ′ _m , L ′ _m , L ′ _m−1 ,..., L ′ ₂ , L ′ ₁ , A ^(u) The order between the matrices is set so as to be in order, and the matrix product calculation unit 71 is controlled to perform matrix multiplication in this order. The matrix multiplication number reduction unit 75, in each multiplication, the number of non-zero elements in the off-diagonal terms of the two matrices to be multiplied and the number of non-zero elements in the off-diagonal terms of one matrix C that is the multiplication result When the difference between the latter and the former is equal to or smaller than a predetermined threshold a (a> 0), two matrices to be multiplied are replaced with one matrix C. When the matrix C is replaced in this way, the matrix C is multiplied by the next matrix.

On the other hand, when the difference between the latter and the former exceeds a predetermined threshold value a, the matrix multiplication number reduction unit 75 does not replace the two matrices to be multiplied with one matrix C, but instead replaces the matrix E ^(v) , U ′ ₁ , U ′ ₂ ,..., U ′ _m−1 , U ′ _m , L ′ _m , L ′ _m−1 ,..., L ′ ₂ , L ′ ₁ , A ^(u) The sum of the number of non-zero elements of the diagonal term and the matrix E ^(v) , U ′ ₁ , U ′ ₂ ,..., U ′ _m−1 , U ′ _m , L ′ _m , L ′ _m−1 , .., L ′ ₂ , L ′ ₁ , A ^(u) excluding the replaced matrix and the sum of the number of non-zero elements in the non-diagonal terms of the matrix C are compared. , U ′ _m−1 , U ′ _m , L ′ _m , L ′ _m−1 , matrix E ^(v) , U ′ ₁ , U ′ ₂ ,. .., L ′ ₂ , L ′ ₁ , A ^(u) except for the replaced matrix and the matrix C are stored in the matrix storage unit 78. When the matrix up to the t-th rank is replaced, the matrix C, U ′ _t , U ′ _{t + 1} ,..., U ′ _m−1 , U ′ _m , L ′ _m , L ′ _m−1 ,..., L ′ ₂ , L ′ ₁ , A ^(u) are stored in the matrix storage unit 78.

In order to calculate the parity vector Y, the matrix product calculation control unit 127 determines that the order of the matrices in which data is stored in the matrix storage unit 78 is A ^(u) , L ′ ₁ , L ′ ₂ ,. , L ′ _m−1 , L ′ _m , U ′ _m , U ′ _m−1 ,..., U ′ _t , C are set in order, and matrix multiplication is performed in this order. The matrix product calculation unit 71 is controlled to perform this. The matrix product calculation control unit 127 causes the matrix product calculation unit 71 to output a parity vector Y as a calculation result after the (2m−t + 3) times of matrix multiplication included in the equation (H28).

(Matrix decomposition processing)
FIG. 21 is a flowchart of matrix decomposition processing according to the fourth embodiment.

Referring to FIG. 21, steps S501 to S504 of the second embodiment are the check matrix decomposition unit 30, the matrix storage unit 78, the intermediate result storage unit 72, the ordering unit 40, the LU decomposition unit 34, and the basic matrix decomposition unit. 36. However, the generated matrix _{^{E (v), U '1}} , U' 2, ···, U 'm-1, U' m, L 'm, L' m-1, ···, L '2 , L ′ ₁ , A ^(u) are stored not in the matrix storage unit 78 but in the intermediate result storage unit 72 as in the second embodiment. (Step S1301).

Next, the matrix multiplication number reduction unit 75 performs E ^(v) , U ′ ₁ , U ′ ₂ ,..., U ′ _m−1 from the top of the matrix for which data is stored in the intermediate result storage unit 72. , U ′ _m , L ′ _m , L ′ _m−1 ,..., L ′ ₂ , L ′ ₁ , A ^(u) are set in this order (step S1302).

Next, the matrix multiplication number reduction unit 75 sets the control variable t to 1 (step S1303).
Next, the matrix multiplication number reduction unit 75 reads data representing the matrix E ^(v) of the t (= 1) th rank from the matrix storage unit 78 and represents the matrix C with the matrix E ^(v) as the matrix C. The data is stored in the intermediate result storage unit 72 (step S1304).

  The matrix number product reduction unit reads the data representing the matrix C from the intermediate result storage unit 72, counts the non-zero element number SC0 of the matrix C, and stores the non-zero element number SC0 in the intermediate result storage unit 72 (step). S1305).

  Next, the matrix multiplication number reduction unit 75 reads data representing the (t + 1) -th rank matrix and sets the matrix as the matrix mat (step S1306).

  Next, the matrix multiplication number reduction unit 75 counts the non-zero element number SC1 of the non-diagonal term of the matrix mat, and the non-zero element number SC1 and the non-zero element number SC0 of the matrix C in the intermediate result storage unit 72 And the sum SC2 of the number of non-zero elements is stored in the intermediate result storage unit 72 (step S1307).

  Next, the matrix product calculation unit 71 calculates the product C × Mat of the matrix C and the matrix Mat, and sets the calculation result as the matrix D (step S1308).

  Next, the matrix multiplication number reduction unit 75 counts the non-zero element number SC3 of the non-diagonal term of the matrix D, and stores the non-zero element number SC3 in the intermediate result storage unit 72 (step S1309).

  The matrix multiplication number reduction unit 75 reads the sum SC2 of the number of non-zero elements and the number of non-zero elements SC3 from the intermediate result storage unit 72, and determines whether the difference between SC3 and SC2 is less than or equal to the threshold value a (step). S1310).

  If the difference between SC3 and SC2 is equal to or less than the threshold value a (YES in step S1310), the matrix multiplication number reduction unit 75 increments the control variable t (step S1311), and the matrix C in the intermediate result storage unit 72 Is updated with the matrix D (step S1312), the number of nonzero elements SC0 in the intermediate result storage unit 72 is updated with the number of nonzero elements SC3, and the processing from step S1306 is repeated.

On the other hand, when the difference between SC3 and SC2 exceeds the threshold value a (NO in step S1310) and the control variable t is 2 or more (YES in step S1314), the matrix multiplication number reduction unit 75 performs a matrix storage unit 78. matrix data are stored in _{^{E (v), U '1}} , U' 2, ···, U 'm-1, U' m, L 'm, L' m-1, ···, L The number of non-zero elements of non-diagonal terms is counted for each of ′ ₂ , L ′ ₁ , and A ^(u) , and the sum SC4 of these non-zero elements is stored in the intermediate result storage unit 72 (step S1315).

Next, the matrix multiplication number reduction unit 75 generates a matrix E ^(v) , U ′ ₁ , U ′ ₂ ,..., U ′ _m−1 , U ′ _m , L ′ _m , L ′ _m−1,. .. Counting the number of non-zero elements of non-diagonal terms for each of the matrices after the (t + 1) -th rank among L ′ ₂ , L ′ ₁ , and A ^(u) , and summing the number of these non-zero elements The sum SC6 of SC5 and the number of nonzero elements SC0 of the matrix C in the intermediate result storage unit 72 is calculated, and the sum SC6 of the number of nonzero elements is stored in the intermediate result storage unit 72 (step S1316).

  The matrix multiplication number reduction unit 75 reads the sum SC4 of the number of nonzero elements and the sum SC6 of the number of nonzero elements from the intermediate result storage unit 72, and determines whether SC6 is SC4 or less (step S1317).

When SC6 is equal to or lower than SC4 (YES in step S1317), the matrix multiplication number reduction unit 75 determines that the matrix E ^(v) , U ′ ₁ , U ′ ₂ ,. ′ M ₋₁ , U ′ _m , L ′ _m , L ′ _m−1 ,..., L ′ ₂ , L ′ ₁ , A ^(u) except for the matrix from the first rank to the t rank matrix _{U 't, U' t +} 1, ···, U 'm-1, U' m, L 'm, L' m-1, ···, L '2, L' 1, A ( ^u) and the matrix C are stored in the matrix storage unit 78 (step S1318).

(Encoded vector generation processing)
FIG. 22 is a flowchart of an encoded vector generation process according to the fourth embodiment.

Referring to FIG. 22, the matrix product calculation control unit 127 determines that the order of the matrices in which data is stored in the matrix storage unit 78 is A ^(u) , L ′ ₁ , L ′ ₂ ,. The order between the matrices is set so as to be in the order of ′ _m , U ′ _m , U ′ _m−1 ,..., U ′ _t , C (step S1401).

  Next, the matrix product calculation control unit 127 sets the control variable i to 1 (step S1402).

The matrix product calculation control unit 127 instructs the matrix product calculation unit 71 to read out the data representing the i th (= 1) -th order matrix A ^(u) from the matrix storage unit 78, and the matrix A ^(u) And the product A ^(u) X of the input information vector X is calculated. The matrix product calculation unit 71 sets the vector that is the multiplication result as the intermediate result R, and stores the data representing the intermediate result R in the intermediate result storage unit 72 (step S1403).

  Next, the matrix product calculation control unit 127 increments the control variable i (step S1404).

When the control variable i is less than (2m−t + 3) (YES in step S1405), the matrix product calculation control unit 127 instructs the matrix product calculation unit 71 to receive the i-th rank matrix (from the matrix storage unit 78). causes read data representing this and Mat _i), by reading the data representing the intermediate results R stored in the intermediate result storage unit 72, the product Mat _i the matrix Mat _i and an intermediate result R XR is calculated. The matrix product calculation unit 71 sets the vector that is the multiplication result as a new intermediate result R, and updates data representing the intermediate result R in the intermediate result storage unit 72 (step S1406). Thereafter, the processing from step S1404 is repeated.

When the control variable i becomes (2m−t + 3) (NO in step S1405), the matrix product calculation control unit 127 instructs the matrix product calculation unit 71 to receive the i-th rank matrix from the matrix storage unit 78. Data representing (Mat _i ) is read out, and data representing the intermediate result R stored in the intermediate result storage unit 72 is read out to obtain a product Mat _i × R of the matrix Mat _i and the intermediate result R. Let it be calculated. The matrix product calculation unit 71 outputs the vector that is the multiplication result as the parity vector Y (step S1407).

  After that, the array unit 20 generates and outputs an n-bit encoded vector Z composed of a k-bit information vector X and an m-bit parity vector Y (step S1408).

(Effect of the fourth embodiment)
As described above, according to the fourth embodiment, the parity vector Y is calculated according to the equation (H28) without performing backward substitution during the encoded vector generation process. Encoding can be performed. In particular, since the matrix B is ordered, the basic upper triangular matrix U ′ _i (i = t to m) and the basic lower triangular matrix L ′ _i (i = 1 to m) are represented by the basic upper triangle of the third embodiment. The matrix U _i (i = t + 1 to m) and the basic lower triangular matrix L _i (i = 1 to m) have a smaller number of non-zero elements, and the calculation amount of the expression (H28) is expressed by the expression ( F14) can be further reduced. In Expression (H28), E ^(v) and U ′ _{1 to} U ′ _t−1 in Expression (D26) of the second embodiment are represented by one matrix C, and the non-diagonal terms of the matrix C Since the number of non-zero elements is equal to or less than the sum of the number of non-zero elements in the off-diagonal terms of the matrix E ^(v) , U ′ _{1 to} U ′ _t−1 , the calculation of Expression (H28) is performed in the second embodiment. The calculation amount is smaller than the calculation of the equation (D26). Further, since the parity vector Y is calculated by sequentially executing the matrix multiplication on the right side of the equation (H28), the result of each matrix multiplication becomes a vector, and the capacity for storing the intermediate result can be reduced.

Further, at the time of matrix decomposition processing, without performing an operation with a large amount of calculation of calculating an inverse matrix of the matrix B, generation of matrices B ′, A ^(u) and E ^(v) by ordering of the matrix B, matrix B LU decomposition of LU ', decomposition of LU decomposed upper triangular matrix into product of basic upper triangular matrix, decomposition of LU decomposed lower triangular matrix into product of basic lower triangular matrix, reduction processing of matrix multiplication number Therefore, it is possible to calculate a matrix necessary for generating the parity vector Y with a small calculation amount.

[Modification 1 of Fourth Embodiment]
The encoder of the first modification of the fourth embodiment performs a real number operation.

(Encoding algorithm)
The encoder receives an information vector X composed of k information bits x1, x2,..., Xk and performs a real number operation to thereby obtain m parity bits y1, y2 satisfying the following expression. ,..., Ym to generate a parity vector Y.
H (X, Y) ^T = 0 (I1)
Here, H is a parity check matrix (hereinafter also referred to as a check matrix), which is a matrix of m rows and n columns (= k + m).
(X, Y) ^T represents transposition of a vector composed of an information vector and a parity vector.

Formula (I1) is modified as follows in the case of real number calculation.
BY = −AX (I2)
Here, the matrix A is a matrix of m rows and k columns including the first to kth columns of the check matrix H. The matrix B is a matrix of m rows and m columns including the (k + 1) -th column to the n-th column of the parity check matrix H.

Assume that ordering is performed on the matrix B as follows.
First, when the two rows of the matrix B are exchanged, the formula (I2) is transformed as follows.
P ₁ BY = P ₁ AX (I3)
Here, P ₁ is a transformation matrix representing the first row replacement. In the case of converting i row and j row, P ₁ is a matrix in which the i row and j row of the unit matrix E are exchanged.

Further, when the two columns of the matrix B are exchanged, the formula (I3) is transformed as follows.
P ₁ BQ ₁ Q ₁ ^-1 Y = -P ₁ AX (I4)
Here, Q ₁ is a transformation matrix representing the first column replacement. When converting the i column and the j column, Q ₁ is a matrix in which the i column and the j column of the unit matrix E are exchanged.

Define a new matrix as follows:
B ⁽¹⁾ = P ₁ BQ ₁ (I5)
A ⁽¹⁾ = P ₁ A (I6)
From the formula (I5) and the formula (I6), the formula (I4) is represented by the following formula.
B ⁽¹⁾ Q ₁ ⁻¹ Y = −A ⁽¹⁾ X (I7)

Further, when two rows of the matrix B ⁽¹⁾ are exchanged, the formula (I7) is transformed as follows.
P ₂ B ⁽¹⁾ Q ₁ ^-1 Y = -P ₂ A ⁽¹⁾ X (I8)
Here, P ₂ is a transformation matrix representing the second row replacement.

Further, when the two columns of the matrix B ⁽¹⁾ are exchanged, the formula (I8) is transformed as follows.
P ₂ B ⁽¹⁾ Q ₂ Q ₂ ^-1 Q ₁ ^-1 Y = -P ₂ A ⁽¹⁾ X (I9)
Here, Q ₂ is a transformation matrix representing the second column replacement.

Define a new matrix as follows:
B ⁽²⁾ = P ₂ B ⁽¹⁾ Q ₂ ... (I10)
A ⁽²⁾ = P ₂ A ⁽¹⁾ ... (I11)
From formula (I10) and formula (I11), formula (I9) is represented by the following formula.
B ⁽²⁾ Q ₂ ^-1 Q ₁ ^-1 Y = -A ⁽²⁾ X (I12)

Similarly, for the matrix B, the row replacement and the v-th (v ≧ 1) extra column are performed for the u-th (u ≧ 1). Assuming that the matrix representing the s-th (1 ≦ s ≦ u) row replacement is P _s and the matrix representing the s-th (1 ≦ s ≦ v) column replacement is Q _s , the following new matrix Define
_{B '= P u ··· P 2} P 1 BQ 1 Q 2 ··· Q v ··· (I13)
A ^(u) = P _u ... P ₂ P ₁ A (I14)
E ^(v) = Q ₁ Q ₂ ... Q _v (I15)
The following formulas are established by formula (I13), formula (I14), and formula (I15).
B'E ^{(v) -1} Y = -A ^(u) X (I16)

When the matrix B ′ is subjected to LU decomposition, the equation (I16) is transformed into the following equation.
L′ U′Q _V ⁻¹ ... Q ₂ ⁻¹ Q ₁ ⁻¹ Y = A ^(u) X (I17)
Here, L ′ is a lower triangular matrix and U ′ is an upper triangular matrix.
From the equation (I17), the column vector Y is expressed by the following equation.
Y = -E ^(v) U'- ¹ L'- ¹ A ^(u) X (I18)

The upper triangular matrix U ′ is decomposed into products of basic upper triangular matrices U ′ i (i = 1 to m) as in the following equation.
_{U '= U' m U '} m-1 ··· U' 2 U '1 ··· (I19)
Here, in the inverse matrix U ′ _i ⁻¹ , the element of the diagonal term is 1, and the element of the off-diagonal term is obtained by inverting the sign of the element of U ′ _i .
From the formula (I19), U ′ ⁻¹ is represented by the following formula.
U ′ ⁻¹ = U ′ ₁ ⁻¹ U ′ ₂ ⁻¹ ... U ′ _m−1 ⁻¹ U ′ _m ⁻¹ (I20)

The lower triangular matrix L is decomposed into products of basic lower triangular matrices L _i (i = 1 to m) as shown in the following expression.
L ′ = L ′ ₁ L ′ ₂ ... L ′ _m−1 L ′ _m (I21)
Here, in the inverse matrix L ′ _i ⁻¹ , the element of the diagonal term is 1, and the element of the non-diagonal term is obtained by inverting the sign of the element of L ′ _i .
From the formula (I21), L ′ ⁻¹ is represented by the following formula.
L ′ ⁻¹ = L ′ _m ⁻¹ L ′ _m−1 ⁻¹ ... L ′ ₂ ⁻¹ L ′ ₁ ⁻¹ (I22)

Formula (I18) is transformed into the following formula by Formula (I20) and Formula (I22).
^{Y = -E (v) U '} 1 -1 U' 2 -1 ··· U 'm-1 -1 U' m -1 L 'm -1 L' m-1 -1 ··· L '2 ^-1 L ' ₁ ^-1 A ^(u) X (I23)

Equation (I23) is a multiplication of (2 × m + 3) matrices, but if certain conditions are satisfied, the product of several matrices can be represented by one matrix. It is assumed that the product of t matrices of E ^(v) and U ′ ₁ ^{−1 to} U ′ _t−1 ⁻¹ can be represented by one matrix C as in the following equation.
E ^(v) (U ′ ₁ ⁻¹ U ′ ₂ ⁻¹ ... U ′ _t−1 ⁻¹ ) ≡C (I24)
In this case, the certain conditions, the matrix _{^{E (v), U '1}} -1, U' 2 -1, ···, U 't-1 of the non-diagonal terms of ^-1 of the number of non-zero elements Assume that the sum is equal to or greater than the number of non-zero elements in the off-diagonal terms of matrix C.

From formula (I24), formula (I23) is expressed by the following formula.
Y = −CU ′ _t ⁻¹ ... U ′ _m−1 ⁻¹ U ′ _m ⁻¹ L ′ _m ⁻¹ L ′ _m−1 ⁻¹ ... L ′ ₂ ⁻¹ L ′ ₁ ⁻¹ A ^{( u)} X (I25)

Using the parity vector Y calculated by the equation (I25) and the information vector X, the encoded vector Z is expressed by the following equation.
Z = (X, Y) (I26)
Here, the first bit to the k-th bit of the encoded vector Z are bits of the information vector X, and the (k + 1) -th bit to the n-th bit of the encoded vector Z are bits of the parity vector Y.

(Constitution)
FIG. 23 is a diagram illustrating a configuration of an encoder according to Modification 1 of the fourth embodiment.

  Referring to FIG. 23, this encoder 2h includes a parity check matrix decomposing unit 30, an ordering unit 40, an LU decomposing unit 34, and a basic matrix decomposing unit 36, which are the same as in the first modification of the second embodiment. And an inverse matrix calculation unit 43, a sign inversion unit 99, and an arrangement unit 20. Description of these will not be repeated.

  The encoder 2h includes an intermediate result storage unit 82, a matrix storage unit 88, a matrix product calculation unit 81, a matrix product calculation control unit 128, and a matrix multiplication number reduction unit 85. In the following description, it is assumed that t <m.

  The intermediate result storage unit 82 stores data representing an intermediate result in a process (matrix decomposition process) for generating a matrix included in the formula (I25). Further, the intermediate result storage unit 82 stores data representing an intermediate result of (2m−t + 3) times of matrix multiplication included in the equation (I25) at the time of generating the parity vector Y.

The matrix storage unit 88 includes data representing the inverse matrices U ′ _t ⁻¹ , U ′ _{t + 1} ⁻¹ ,..., U ′ _m ⁻¹ of the basic upper triangular matrix, and the inverse matrix L ′ _m of the basic lower triangular matrix. ⁻¹ , L ′ _m−1 ⁻¹ ,..., L ′ ₁ ⁻¹ , and data representing the matrices C and A ^(u) are stored.

The matrix product calculation unit 81 includes an adder 87 and a multiplier 89. In order to calculate the parity vector Y, the matrix product calculation unit 81 performs (2m−t + 3) times of matrix multiplications included in the equation (I25) in the order instructed by the matrix product calculation control unit 128. The matrix product calculation unit 81 inputs the input information vector X, the matrix A ^(u) , L ′ ₁ ⁻¹ , L ′ ₂ ⁻¹ ,..., L ′ _m−1 ⁻¹ in the matrix storage unit 88. Among data representing L ′ _m ⁻¹ , U ′ _m ⁻¹ , U ′ _m−1 ⁻¹ ,..., U ′ _t ⁻¹ , C, and data representing intermediate results in the intermediate result storage unit 82 Are used to perform each matrix multiplication. The matrix product calculation unit 81 stores the data representing the calculation result in the intermediate result storage unit 82 as the data representing the intermediate result until (2m−t + 3) times of matrix multiplication included in the formula (I25) is completed. Let Further, the matrix product calculation unit 81 executes the matrix multiplication instructed by the matrix multiplication number reduction unit 85 when executing the process of reducing the number of matrix multiplications included in the formula (I23).

When the matrix multiplication number reduction unit 85 performs the process of reducing the number of matrix multiplications included in the formula (I23), the order of the matrices in which the data is stored in the intermediate result storage unit 82 is E ^(v) , U ′ ₁ ⁻¹ , U ′ ₂ ⁻¹ ,..., U ′ _m−1 ⁻¹ , U ′ _m ⁻¹ , L ′ _m ⁻¹ , L ′ _m−1 ⁻¹ ,. The order between the matrices is set so as to be in the order of ' ₂ ^-1 , L' ₁ ^-1 , A ^(u) , and the matrix product calculation unit 81 is controlled so as to perform matrix multiplication in this order. The matrix multiplication number reduction unit 85, in each multiplication, the number of non-zero elements in the off-diagonal terms of the two matrices to be multiplied and the number of non-zero elements in the off-diagonal terms of one matrix C that is the multiplication result When the difference between the latter and the former is equal to or smaller than a predetermined threshold a (a> 0), two matrices to be multiplied are replaced with one matrix C. When the matrix C is replaced in this way, the matrix C is multiplied by the next matrix.

On the other hand, when the difference between the latter and the former exceeds a predetermined threshold value a, the matrix multiplication number reduction unit 85 does not replace the two matrices to be multiplied with one matrix C, but instead replaces the matrix E ^(v) , U ′ ₁ ⁻¹ , U ′ ₂ ⁻¹ ,..., U ′ _m−1 ⁻¹ , U ′ _m ⁻¹ , L ′ _m ⁻¹ , L ′ _m−1 ^−1,. ₂ ⁻¹ , L ′ ₁ ⁻¹ , the sum of the number of non-zero elements in the off-diagonal terms of A ^(u) and the matrix E ^(v) , U ′ ₁ ⁻¹ , U ′ ₂ ⁻¹ ,. Of U ′ _m−1 ⁻¹ , U ′ _m ⁻¹ , L ′ _m ⁻¹ , L ′ _m−1 ⁻¹ ,..., L ′ ₂ ⁻¹ , L ′ ₁ ⁻¹ , A ^(u) Are compared with the sum of the number of non-zero elements in the off-diagonal term of the matrix C. If the latter value is less than or equal to the former value, the matrix ^v) , U ′ _{^{1 -1, U '2 -1,}} ···, U' m-1 -1, U 'm -1, L' m -1, L 'm-1 -1, ···, L' 2 - ¹ , L ′ ₁ ⁻¹ , matrix excluding replaced matrix of A ^(u) And the matrix C are stored in the matrix storage unit 88. When the matrix up to the t-th rank is replaced, the matrix C, U ′ _t ⁻¹ , U ′ _{t + 1} ⁻¹ ,..., U ′ _m−1 ⁻¹ , U ′ _m ⁻¹ , L _{^{'m -1, L' m-}} 1 -1, ···, L '2 -1, L' 1 -1, a (u) is stored in the matrix storage unit 88.

In order to calculate the parity vector Y, the matrix product calculation control unit 128 determines that the order of the matrices in which data is stored in the matrix storage unit 88 is A ^(u) , L ′ ₁ ⁻¹ , L ′ ₂ ^{−1 from the top.} _{, ···, L 'm-1} -1, L' m -1, U 'm -1, U' m-1 -1, ···, U 't -1, such that the order of C The order between the matrices is set, and the matrix product calculation unit 81 is controlled to perform matrix multiplication in this order. Further, the matrix product calculation control unit 128, after the completion of (2m−t + 3) matrix multiplications included in the formula (I25), the vector Y ′ (= CU ′ _t ^{−1) as the} calculation result from the matrix product calculation unit 81.・ ・ ・ U ' _m-1 ^-1 U' _m ^-1 L ' _m ^-1 L' _m-1 ^-1 ... L ' ₂ ^-1 L' ₁ ^-1 A ^(u) X) Output to 99.

(Matrix decomposition processing)
FIG. 24 is a flowchart of matrix decomposition processing according to Modification 1 of the fourth embodiment.

Referring to FIG. 24, first, Steps S701 to S705 of Modification 1 of the second embodiment are the check matrix decomposition unit 30, the matrix storage unit 88, the intermediate result storage unit 82, the ordering unit 40, and the LU decomposition unit 34. This is executed by the basic matrix decomposition unit 36 and the inverse matrix calculation unit 43. However, the generated matrices E ^(v) , U ′ ₁ ⁻¹ , U ′ ₂ ⁻¹ ,..., U ′ _m−1 ⁻¹ , U ′ _m ⁻¹ , L ′ _m ⁻¹ , L ′ _{m −1} ⁻¹ ,..., L ′ ₂ ⁻¹ , L ′ ₁ ⁻¹ , A ^(u) are not stored in the matrix storage unit 88 as in the first modification of the second embodiment, but are stored as intermediate results Stored in the unit 82 (step S1501).

Next, the matrix multiplication number reduction unit 85 applies E ^(v) , U ′ ₁ ⁻¹ , U ′ ₂ ⁻¹ ,..., U from the top of the matrix for which data is stored in the intermediate result storage unit 82. ′ M ₋₁ ⁻¹ , U ′ _m ⁻¹ , L ′ _m ⁻¹ , L ′ _m−1 ⁻¹ ,..., L ′ ₂ ⁻¹ , L ′ ₁ ⁻¹ , A ^(u) The order is set so as to be (step S1502).

Next, the matrix multiplication number reduction unit 85 sets the control variable t to 1 (step S1503).
Next, the matrix multiplication number reduction unit 85 reads data representing the t (= 1) -th rank matrix E ^(v) from the matrix storage unit 88 and represents the matrix C with the matrix E ^(v) as the matrix C. Data is stored in the intermediate result storage unit 82 (step S1504).

  The matrix number product reduction unit reads the data representing the matrix C from the intermediate result storage unit 82, counts the non-zero element number SC0 of the matrix C, and stores the non-zero element number SC0 in the intermediate result storage unit 82 (step). S1505).

  Next, the matrix multiplication number reduction unit 85 reads data representing the (t + 1) -th rank matrix and sets the matrix as the matrix mat (step S1506).

  Next, the matrix multiplication number reduction unit 85 counts the number of non-zero elements SC1 of the non-diagonal term of the matrix mat, and the number of non-zero elements SC1 and the number of non-zero elements SC0 of the matrix C in the intermediate result storage unit 82 And SC2 of the number of non-zero elements is stored in the intermediate result storage unit 82 (step S1507).

  Next, the matrix product calculation unit 81 calculates a product C × Mat of the matrix C and the matrix Mat, and sets the calculation result as the matrix D (step S1508).

  Next, the matrix multiplication number reduction unit 85 counts the non-zero element number SC3 of the non-diagonal term of the matrix D, and stores the non-zero element number SC3 in the intermediate result storage unit 82 (step S1509).

  The matrix multiplication number reduction unit 85 reads the sum SC2 of the number of non-zero elements and the number of non-zero elements SC3 from the intermediate result storage unit 82, and determines whether the difference between SC3 and SC2 is less than or equal to the threshold value a (step S1510).

  If the difference between SC3 and SC2 is equal to or smaller than the threshold value a (YES in step S1510), the matrix multiplication number reduction unit 85 increments the control variable t (step S1511), and the matrix C in the intermediate result storage unit 82 Is updated with the matrix D (step S1512), the number of nonzero elements SC0 in the intermediate result storage unit 82 is updated with the number of nonzero elements SC3, and the processing from step S1506 is repeated.

On the other hand, when the difference between SC3 and SC2 exceeds the threshold value a (NO in step S1510) and the control variable t is 2 or more (YES in step S1514), the matrix multiplication unit reduction unit 85 matrix processing unit 88 matrix data are stored in _{^{E (v), U '1}} -1, U' 2 -1, ···, U 'm-1 -1, U' m -1, L 'm -1, L ′ M ₋₁ ⁻¹ ,..., L ′ ₂ ⁻¹ , L ′ ₁ ⁻¹ , and A ^(u) are counted for the number of non-diagonal non-zero elements, The sum SC4 is stored in the intermediate result storage unit 82 (step S1515).

Next, the matrix multiplication number reduction unit 85 performs the matrix E ^(v) , U ′ ₁ ⁻¹ , U ′ ₂ ⁻¹ ,..., U ′ _m−1 ⁻¹ , U ′ _m ⁻¹ , L ′ _m ⁻¹ , L ′ _m−1 ⁻¹ ,..., L ′ ₂ ⁻¹ , L ′ ₁ ⁻¹ , A ^(u) , the matrix after the (t + 1) th rank The number of non-zero elements is counted, and the sum SC6 of the total number SC5 of these non-zero elements and the number of non-zero elements SC0 of the matrix C in the intermediate result storage unit 82 is calculated. The result is stored in the intermediate result storage unit 82 (step S1516).

  The matrix multiplication number reduction unit 85 reads the sum SC4 of the number of nonzero elements and the sum SC6 of the number of nonzero elements from the intermediate result storage unit 82, and determines whether SC6 is SC4 or less (step S1517).

When SC6 is equal to or lower than SC4 (YES in step S1517), the matrix multiplication number reduction unit 85 determines that the matrix E ^(v) , U ′ ₁ ⁻¹ , U ′ ₂ ⁻¹ ,. .., U ' _m-1 ^-1 , U' _m ^-1 , L ' _m ^-1 , L' _m-1 ^-1 , ..., L ' ₂ ^-1 , L' ₁ ^-1 , A ^{(u )} , U ′ _t ⁻¹ , U ′ _{t + 1} ⁻¹ ,..., U ′ _m−1 ⁻¹ , U ′ _m ⁻¹ , excluding the first to t-th order matrices L ′ _m ⁻¹ , L ′ _m−1 ⁻¹ ,..., L ′ ₂ ⁻¹ , L ′ ₁ ⁻¹ , A ^(u) and the matrix C are stored in the matrix storage unit 88 (step S1518). ).

(Encoded vector generation processing)
FIG. 25 is a flowchart of an encoded vector generation process according to Modification 1 of the fourth embodiment.

Referring to FIG. 25, the matrix product calculation control unit 128 determines that the order of the matrices in which data is stored in the matrix storage unit 88 is A ^(u) , L ′ ₁ ⁻¹ , L ′ ₂ ^−1,. .., L ′ _m ⁻¹ , U ′ _m ⁻¹ , U ′ _m−1 ⁻¹ ,..., U ′ _t ⁻¹ , C are set in the order of the matrix (step S1601). ).

  Next, the matrix product calculation control unit 128 sets the control variable i to 1 (step S1602).

The matrix product calculation control unit 128 instructs the matrix product calculation unit 81 to read out the data representing the i th (= 1) -th order matrix A ^(u) from the matrix storage unit 88, and the matrix A ^(u) And the product A ^(u) X of the input information vector X is calculated. The matrix product calculation unit 81 sets the vector that is the multiplication result as the intermediate result R, and stores the data representing the intermediate result R in the intermediate result storage unit 82 (step S1603).

  Next, the matrix product calculation control unit 128 increments the control variable i (step S1604).

If the control variable i is less than (2m−t + 3) (YES in step S1605), the matrix product calculation control unit 128 instructs the matrix product calculation unit 81 to receive the i-th rank matrix (from the matrix storage unit 88). causes read data representing this and Mat _i), by reading the data representing the intermediate results R stored in the intermediate result storage unit 82, the product Mat _i the matrix Mat _i and an intermediate result R XR is calculated. The matrix product calculation unit 81 sets the vector that is the multiplication result as a new intermediate result R, and updates the data representing the intermediate result R in the intermediate result storage unit 82 (step S1606). Thereafter, the processing from step S1604 is repeated.

When the control variable i becomes (2m−t + 3) (NO in step S1605), the matrix product calculation control unit 128 instructs the matrix product calculation unit 81 to receive the i-th rank matrix from the matrix storage unit 88. Data representing (Mat _i ) is read out, and data representing the intermediate result R stored in the intermediate result storage unit 82 is read out to obtain a product Mat _i × R of the matrix Mat _i and the intermediate result R. Let it be calculated. The matrix product calculation unit 81 outputs a vector Y ′ as a multiplication result to the sign inversion unit 99. The sign reversing unit 99 outputs the inverted vector Y ′ as the parity vector Y (step S1607).

  After that, the array unit 20 generates and outputs an n-bit encoded vector Z composed of a k-bit information vector X and an m-bit parity vector Y (step S1608).

(Effect of Modification 1 of Fourth Embodiment)
As described above, according to the first modification of the fourth embodiment, the parity vector Y is calculated according to the equation (I25) without performing backward substitution at the time of the encoded vector generation process. The information vector X can be encoded. In particular, since the matrix B is ordered, the inverse matrix U ′ _i ⁻¹ (i = t to m) of the basic upper triangular matrix and the inverse matrix L ′ _i ⁻¹ (i = ^{1 to} m) of the basic lower triangular matrix are The inverse upper matrix U _i ⁻¹ (i = t + ^{1 to} m) of the basic upper triangular matrix and the inverse matrix L _i ⁻¹ (i = ^{1 to} m) of the basic lower triangular matrix of the first modification of the third embodiment Thus, the number of non-zero elements is smaller, and the amount of calculation of the equation (I25) can be further reduced than that of the equation (G11). In Expression (I25), E ^(v) and U ′ ₁ ^{−1 to} U ′ _t−1 ⁻¹ in Expression (E23) of Modification 1 of the second embodiment are represented by one matrix C, and Since the number of non-zero elements of the non-diagonal terms of the matrix C is less than or equal to the sum of the number of non-zero elements of the non-diagonal terms of the matrix E ^(v) , U ′ ₁ ^{−1 to} U ′ _t−1 ⁻¹ The calculation of (I25) has a smaller amount of calculation than the calculation of Expression (E23) in Modification 1 of the second embodiment. Further, since the parity vector Y is calculated by sequentially executing the matrix multiplication on the right side of the formula (I25), the result of each matrix multiplication becomes a vector, and the capacity for storing the intermediate result can be reduced.

Further, at the time of matrix decomposition processing, without performing an operation with a large amount of calculation of calculating an inverse matrix of the matrix B, generation of matrices B ′, A ^(u) and E ^(v) by ordering of the matrix B, matrix B LU decomposition of ′, decomposition of LU decomposed upper triangular matrix into product of basic upper triangular matrix, decomposition of LU decomposed lower triangular matrix into product of basic lower triangular matrix, basic upper triangular matrix and basic lower triangular matrix Therefore, the matrix necessary for generating the parity vector Y can be calculated with a small amount of calculation.

[Fifth Embodiment]
The encoder of the fifth embodiment performs an operation modulo 2. This encoder performs encoded vector processing in parallel by a plurality of matrix product calculation units.

(Encoding algorithm)
The encoding algorithm of the fifth embodiment is the same as that of the first embodiment.

(Constitution)
FIG. 26 is a diagram illustrating a configuration of an encoder according to the fifth embodiment.

  Referring to FIG. 26, the encoder 2i includes a parity check matrix decomposition unit 10, an LU decomposition unit 14, a basic matrix decomposition unit 16, a matrix storage unit 18, and an array similar to those in the first embodiment. Part 20.

  The encoder further includes an intermediate result storage unit 92, a matrix product calculation unit 91, and a matrix product calculation control unit 129.

  The matrix product calculation unit 91 includes a first matrix product calculation unit 97 and a second matrix product calculation unit 98.

The first matrix product calculation unit 97 includes an adder 94 and a multiplier 93. In order to calculate the parity vector Y, the matrix product calculation unit 91 instructs the matrix product calculation control unit 129 to perform (m + 1) -th matrix multiplication from the first to (m + 1) th from the right included in the equation (B12). Run in the order specified. The matrix product calculation unit 91 receives the input information vector X, data representing the matrices L _m , L _m−1 ,..., L ₁ , A in the matrix storage unit 18, and the intermediate in the intermediate result storage unit 92. Each matrix multiplication is executed using two of the data representing the results, and the data representing the calculation results are stored in the intermediate result storage unit 92 as data representing the intermediate results. When (m + 1) matrix multiplications are completed, a column vector R ₁ (= L _m L _m−1 ... L ₂ L ₁ AX) as an intermediate result is obtained.

The second matrix product calculation unit 98 includes an adder 96 and a multiplier 95. In order to calculate the parity vector Y, the matrix product calculation unit 91 instructs the matrix product calculation control unit 129 to perform m times of matrix multiplication from the (m + 2) th to the (2m + 1) th from the right included in the equation (B12). Run in the order specified. The matrix product calculation unit 91 uses _{two of} data representing the matrices U ₁ , U ₂ ,..., U _m in the matrix storage unit 18 and data representing the intermediate results in the intermediate result storage unit 92. Each matrix multiplication is executed, and data representing the calculation result is stored in the intermediate result storage unit 92 as data representing the intermediate result. When m matrix multiplications are completed, a matrix R ₂ (= U ₁ U ₂ ... U _m−1 U _m ) as an intermediate result is obtained.

The first matrix product calculation unit 97 calculates a product R ₂ × R ₁ of the intermediate result R ₁ and the intermediate result R ₂ and outputs a vector as a calculation result as the parity vector Y.

Matrix product calculation control unit 129, in order to calculate the parity vector Y, A sequence from the beginning of the matrix data in the matrix storage unit 18 is _{stored, L 1, L 2, ···} , L m-1 , L _m , U _m , U _m−1 ,..., U ₂ , U ₁ are set in order, and the matrix product calculation unit 91 is set to perform matrix multiplication in this order. Control. More specifically, the matrix product calculation control unit 129 causes the first matrix multiplier to perform matrix multiplication on the information vector X, the first rank to the (m + 1) th rank matrix in this order. After all matrix multiplications are completed, data representing intermediate result R ₁ is output to first matrix product calculation unit 97. Further, the matrix product calculation control unit 129 causes the second matrix multiplier to perform matrix multiplication on the (m + 1) -th to (2m + 1) -th rank matrices in this order, and all these matrix multiplications are completed. Later, data representing the intermediate result R ₂ is output to the first matrix product calculator 97. Further, the matrix product calculation control unit 129 causes the first matrix product calculation unit 97 to output a parity vector Y as a calculation result after the (2m + 1) times of matrix multiplication included in the equation (B12).

  The intermediate result storage unit 92 stores data representing an intermediate result in the process (matrix decomposition process) for generating the matrix included in the formula (B12). In addition, the intermediate result storage unit 92 includes (2m + 1) times included in the expression (B12) output from the first matrix product calculation unit 97 and the second matrix product calculation unit 98 during the parity vector Y generation process. Data representing intermediate results of matrix multiplication is stored.

(Matrix decomposition processing)
Since the matrix decomposition processing procedure of the fifth embodiment is the same as that of the first embodiment, the description thereof will not be repeated here.

(Encoded vector generation processing)
FIG. 27 is a flowchart of an encoded vector generation process according to the fifth embodiment.

Referring to FIG. 27, the matrix product calculation control unit 129 determines that the order of the matrices in which data is stored in the matrix storage unit 18 is A, L ₁ , L ₂ ,..., L _m−1 , L The order between the matrices is set so as to be in the order of _m , U _m , U _m−1 ,..., U ₂ , U ₁ (step S1701).

  The matrix product calculation control unit 129 performs the processes of steps S1702 to S1706 for the first matrix product calculation unit 97 and the processes of steps S1707 to S1711 for the second matrix product calculation unit 98 simultaneously in parallel. Control.

The matrix product calculation control unit 129 sets the control variable i to 1 (step S1702).
The matrix product calculation control unit 129 instructs the first matrix product calculation unit 97 to read out the data representing the i-th (= 1) -th rank matrix A from the matrix storage unit 18 and input the matrix A and the input The product AX with the calculated information vector X is calculated. The first matrix product calculation unit 97 sets the vector, which is the multiplication result, as the intermediate result R ₁ and stores data representing the intermediate result R ₁ in the intermediate result storage unit 92 (step S1703).

  Next, the matrix product calculation control unit 129 increments the control variable i (step S1704).

When the control variable i is equal to or less than (m + 1) (YES in step S1705), the matrix product calculation control unit 129 instructs the first matrix product calculation unit 97 to transmit the i-th rank matrix from the matrix storage unit 18. (This is referred to as Mat _i ) is read out, and data representing the intermediate result R ₁ stored in the intermediate result storage unit 92 is read out so that the matrix Mat _i and the intermediate result R ₁ are Let the product Mat _i × R ₁ be calculated. The first matrix product calculation unit 97 sets the vector that is the multiplication result as a new intermediate result R _1, and updates data representing the intermediate result R ₁ in the intermediate result storage unit 92 (step S1706). Thereafter, the processing from step S1704 is repeated.

  In addition, the matrix product calculation control unit 129 sets the control variable j to (m + 2) (step S1707).

The matrix product calculation control unit 129 instructs the second matrix product calculation unit 98 to read data representing the matrix U _m of the j (= m + 2) -th rank from the matrix storage unit 18, so that the matrix U _m The intermediate result R ₂ is used, and data representing the intermediate result R ₂ is stored in the intermediate result storage unit 92 (step S1708).

  Next, the matrix product calculation control unit 129 increments the control variable j (step S1709).

If the control variable j is equal to or less than (2m + 1) (YES in step S1710), the matrix product calculation control unit 129 instructs the second matrix product calculation unit 98 to transmit the jth rank matrix from the matrix storage unit 18. (This is referred to as Mat _j ) is read out, and data representing the intermediate result R ₂ stored in the intermediate result storage unit 92 is read out so that the matrix Mat _j and the intermediate result R ₂ are Let the product Mat _j × R ₂ be calculated. The second matrix product calculation unit 98 sets the vector that is the multiplication result as a new intermediate result R ₂ and updates data representing the intermediate result R ₂ in the intermediate result storage unit 92 (step S1711). Thereafter, the processing from step S1709 is repeated.

If the control variable i exceeds (m + 1) (NO in step S1705), the matrix product calculation control unit 129, the data representing the intermediate results R ₁ of the intermediate result storage unit 92 first matrix product calculator When the control variable j exceeds (2m + 1) (NO in step S1710), the matrix product calculation control unit 129 stores the data representing the intermediate result R ₂ in the intermediate result storage unit 92 as the first data. To the matrix product calculation unit 97. The first matrix product calculation unit 97 calculates the product R ₂ × R ₁ of the intermediate result R ₁ and the intermediate result R _2, and outputs a vector as the calculation result as the parity vector Y (step S1712).

  Thereafter, the array unit 20 generates and outputs an n-bit encoded vector Z composed of a k-bit information vector X and an m-bit parity vector Y (step S1713).

(Effect of 5th Embodiment)
As described above, the fifth embodiment has the same effects as those of the first embodiment. Furthermore, according to the fifth embodiment, unlike the backward substitution by sequential calculation in Non-Patent Document 2 and Non-Patent Document 3, the first calculation is performed using the fact that the calculation of Expression (B12) can be executed in parallel. The column vector R ₁ (= L _m L _m−1 ... L ₂ L ₁ AX) calculation processing by the matrix product calculation unit and the matrix R ₂ (= U ₁ U _2. .. U _m−1 U _m ) are simultaneously executed in parallel, so that the parity vector R can be calculated in a shorter time than in Non-Patent Document 2 and Non-Patent Document 3.

[First Modification of Fifth Embodiment]
In embodiments other than the first embodiment, the parity vector generation processing may be executed in parallel by two matrix product calculation units.

For example, in the first modification of the first embodiment, the column vector R ₁ (= L _m ⁻¹ L _m−1 ⁻¹ ... L ₂ ⁻¹ L ₁ ⁻¹ AX) by the first matrix product calculation unit. And the calculation process of the matrix R ₂ (= U ₁ ⁻¹ U ₂ ⁻¹ ... U _m−1 ⁻¹ U _m ⁻¹ ) by the second matrix product calculation unit are executed simultaneously in parallel. The first matrix product calculation unit may calculate the product R ₂ × R ₁ .

In the second embodiment, calculation processing of the column vector R ₁ (= L ′ _m L ′ _m−1 ... L ′ ₂ L ′ ₁ A ^(u) X) by the first matrix product calculation unit, The matrix R ₂ (= E ^(v) U ′ ₁ U ′ ₂ ... U ′ _m−1 U ′ _m ) calculation process by the matrix product calculation unit ₂ is executed in parallel, and the first matrix product The calculation unit may calculate the product R ₂ × R ₁ .

In the first modification of the second embodiment, the column vector R ₁ (= L ′ _m ⁻¹ L ′ _m−1 ⁻¹ ... L ′ ₂ ⁻¹ L ′ ₁ ⁻¹⁾ by the first matrix product calculation unit. A ^(u) X) calculation process and the matrix R ₂ (= E ^(v) U ′ ₁ ⁻¹ U ′ ₂ ⁻¹ ... U ′ _m−1 ⁻¹ U ′ by the second matrix product calculation unit The calculation process of _m ⁻¹ ) may be executed simultaneously in parallel, and the first matrix product calculation unit may calculate the product R ₂ × R ₁ .

In the third embodiment, the column matrix R ₁ (= L _m L _m−1 ... L ₂ L ₁ AX) calculation process by the first matrix product calculation unit and the matrix by the second matrix product calculation unit The calculation process of R ₂ (= CU _{t + 1} ... U _m−1 U _m ) may be executed simultaneously in parallel, and the first matrix product calculation unit may calculate the product R ₂ × R _1. .

In the first modification of the third embodiment, the column vector R ₁ (= L _m ⁻¹ L _m−1 ⁻¹ ... L ₂ ⁻¹ L ₁ ⁻¹ AX) is calculated by the first matrix product calculation unit. The processing and the calculation processing of the matrix R ₂ (= CU _{t + 1} ⁻¹ ... U _m−1 ⁻¹ U _m ⁻¹ ) by the second matrix product calculation unit are executed simultaneously in parallel, and the first The matrix product calculation unit may calculate the product R ₂ × R ₁ .

In the fourth embodiment, the first matrix product calculation unit calculates column vectors R ₁ (= L ′ _m L ′ _m−1 ... L ′ ₂ L ′ ₁ A ^(u) X), The matrix R ₂ (= CU ′ _t ... U ′ _m−1 U ′ _m ) calculation process by the matrix product calculation unit ₂ is executed in parallel, and the first matrix product calculation unit performs the product R ₂ × R ₁ may be calculated.

In the first modification of the fourth embodiment, the column vector R ₁ (= L ′ _m ⁻¹ L ′ _m−1 ⁻¹ ... L ′ ₂ ⁻¹ L ′ ₁ ⁻¹⁾ by the first matrix product calculation unit. A ^(u) X) calculation processing and matrix R ₂ (= CU ′ _t ⁻¹ ... U ′ _m−1 ⁻¹ U ′ _m ⁻¹ ) calculation processing by the second matrix product calculation unit are performed simultaneously. It may be executed in parallel, and the first matrix product calculation unit may calculate the product R ₂ × R ₁ .

[Modification 2 of Fifth Embodiment]
In the fifth embodiment and the first modification of the fifth embodiment, two matrix product calculation units are provided, but parity vector generation processing is executed in parallel by three or more matrix product calculation units. It is good.

(Modification)
The present invention is not limited to the above embodiment, and includes, for example, the following modifications.

(1) Configuration of Encoder In the embodiment of the present invention, the encoder has functions of both parity vector generation processing and matrix decomposition processing, but may have only one function. . When the encoder has only the parity vector generation processing function, the equations (B12), (C9), (D26), (E23), (F14), (G11) for generating the parity vector Y, The matrix data included in (H28) and (I25) is stored in advance as an externally calculated data. For example, in the first embodiment, matrix _{U 1, U 2, ···,} U m, L m, L m-1, ···, a value of each element of L _1, A is calculated in advance outside The calculation results are stored in the matrix storage unit.

(2) Data stored in storage unit In the first to fifth embodiments of the present invention, data representing a matrix or a vector stored in the matrix storage unit and the intermediate result storage unit needs to be a value of each element. For example, it may represent the position of the number of non-zero elements in the off-diagonal term (that is, “1”).

(3) Matrix Product Calculation Order In the embodiment of the present invention, the matrix multiplication on the right side of the matrix multiplication included in the expression for generating the parity vector Y (here, the multiplication of the vector and the matrix is also called matrix multiplication) However, the present invention is not limited to this, and matrix multiplication may be executed in an arbitrary order. However, it may be better not to sequentially execute matrix multiplication on the left side. This is because when the matrix multiplication on the left side is sequentially performed, an inverse matrix of the matrix B is generated, and there is a case where the meaning of decomposing the matrix B is lost. However, even when calculating in this order, matrix decomposition processing can be performed with a smaller amount of calculation than calculating the inverse matrix of the matrix B, or the sum of the number of non-zero elements of the decomposed matrix can be calculated. , B may be less than the sum of the number of non-zero elements of the inverse matrix of B, the capacity for storing the matrix may be reduced.

(4) Matrix Multiplication Number Reduction Process In the third and fourth embodiments of the present invention, the matrix multiplication on the left side of the matrix multiplication included in the expression for generating the parity vector Y is performed during the matrix multiplication number reduction process. Although the sequential execution is performed, the present invention is not limited to this, and matrix multiplication may be performed in an arbitrary order.

For example, the matrix operation may be executed from the right side excluding multiplication with X in the equation for generating the parity vector Y. In this case, instead of the formula (F14), a plurality of adjacent matrices on the right side excluding X in the formula for generating the parity vector Y as shown in the following formula are combined into one matrix C. An equation is used.
Y≡U ₁ ... U _m-1 U _m L _m L _m-1 ... L ₉ CX (F14-1)
Here, C is L ₈ L _t7 ... L ₁ A.

Alternatively, the matrix multiplication at the left end of the expression for generating the parity vector Y and the matrix multiplication at any place other than the right end excluding the multiplication with X may be sequentially executed in any right or left direction. In this case, instead of the equation (F14), a plurality of adjacent matrices other than the left end and the right end (excluding X) of the equation for generating the parity vector Y as in the following equation is a single matrix C. It may be transformed into the form summarized in
Y≡U ₁ U ₂ CU ₉ ... U _m-1 U _m L _m L _m-1 ... L ₂ L ₁ AX (F14-2)
Here, C is U ₃ U ₄ ... U ₈ .

The plurality of adjacent matrices to be replaced with one matrix is not limited to one set, and may be a plurality of sets as in the following equation.
_{Y≡C 1 U 6 U 7 U 8} C 2 U 20 ··· U m-1 U m L m L m-1 ··· L 6 C 3 X ··· (F14-3)
Here, C ₁ is _{U 1 U 2 ··· U 5,} C 2 is the _{U 9 U 10 ··· U 19,} C 3 is a L ₅ L ₄ ··· L ₁ A is there.

  In addition, the sum of the number of non-zero elements in the non-diagonal terms of the matrix included in the formula for generating the parity vector Y in which a plurality of adjacent matrices are replaced with one matrix is the formula for generating the original parity vector Y. When the condition below the sum of the number of non-zero elements of the off-diagonal terms of the included matrix is satisfied, a matrix multiplication number reduction process is performed to combine a plurality of adjacent matrices into one matrix. The matrix multiplication number reduction process may be performed regardless of whether or not the condition is satisfied. That is, the processing of steps S915 to S917 of the third embodiment, the processing of steps S1115 to S1117 of the first modification of the third embodiment, the processing of steps S1315 to S1317 of the fourth embodiment, the fourth embodiment. It is good also as omitting the process of step S1515-S1517 of the modification 1.

  In the embodiment of the present invention, the threshold value a is a positive value, but this is merely an example, and the threshold value a may be 0 or a negative value.

Also, a certain number of adjacent matrices may be unconditionally combined into one matrix.
(5) Ordering In the embodiment of the present invention, ordering is performed on the matrix B by performing u times (u ≧ 1) row replacement and v times (v ≧ 1) column replacement. It is not limited to. For example, only u rows may be replaced (that is, v = 0). In this case, for example, formulas (D26), (E23), (H26), (H28), (I23), and (I25) are transformed into formulas that do not include E ^(v) . Moreover, it is good also as what replaces only the column of v times (that is, u = 0). In this case, for example, formulas (D26), (E23), (H26), (H28), (I23), and (I25) are transformed into formulas that do not include A ^(u) .

  Hereinafter, how the configuration of each unit is modified when the above-described modification is performed in each embodiment will be described.

(5-1) Second Embodiment (5-1-A) u ≧ 1 and v ≧ 1
In the second embodiment, the equation (D26) is used when u ≧ 1 and v ≧ 1.
Y≡E ^(v) U ' ₁ U' ₂ ... U ' _m-1 U' _m L ' _m L' _m-1 ... L ' ₂ L' ₁ A ^(u) X (D26 )

(5-1-B) u ≧ 1 and v = 0
When u ≧ 1 and v = 0, the following equation is used instead of the equation (D26).
Y≡U ′ ₁ U ′ ₂ ... U ′ _m−1 U ′ _m L ′ _m L ′ _m−1 ... L ′ ₂ L ′ ₁ A ^(u) X (D26) ′
Here, the matrix B ′ is obtained by performing u times of row replacement on the matrix B. The matrix L ′ _i (i = 1 to m) is a basic lower triangular matrix of the lower triangular matrix L ′ obtained by LU decomposition of the matrix B ′. The matrix U ′ _i (i = 1 to m) is a basic upper triangular matrix of the upper triangular matrix U ′ obtained by LU decomposition of the matrix B ′.

The ordering unit reads the data representing the matrix A and the matrix B from the intermediate result storage unit, performs the ordering of the matrix B, generates a matrix B ′ in which u rows are exchanged for the matrix B, and generates an intermediate The result is stored in the result storage unit, and a matrix A ^{(u) in} which the same replacement as the u-time replacement of the matrix B is performed on the matrix A is generated and stored in the matrix storage unit.

Matrix storage unit, the basic upper triangular matrix _{U '1, U' 2,} ···, ' data representing the _m, basic lower triangular matrix _{L' U m, L 'm} -1, ···, L' ₁ Data representing and data representing the matrix A ^(u) are stored.

  In order to calculate the parity vector Y, the matrix product calculation unit performs (2m + 1) times of matrix multiplications included in the equation (D26) ′ in the order instructed by the matrix product calculation control unit.

In order to calculate the parity vector Y, the matrix product calculation control unit determines that the order of the matrices in which data is stored in the matrix storage unit is A ^(u) , L ′ ₁ , L ′ ₂ ,. The order between the matrices is set in the order of ′ _m , U ′ _m , U ′ _m−1 ,..., U ′ ₁ , and the matrix product calculation unit is controlled to perform matrix multiplication in this order. . The matrix product calculation control unit outputs a parity vector Y as a calculation result from the matrix product calculation unit after completing (2m + 1) times of matrix multiplication included in the equation (D26) ′.

  The intermediate result storage unit stores data representing an intermediate result in a process (matrix decomposition process) for generating a matrix included in the expression (D26) ′. In addition, the intermediate result storage unit stores data representing the intermediate result of (2m + 1) times of matrix multiplication included in the expression (D26) ′ during the generation process of the parity vector Y.

(5-1-C) u = 0 and v ≧ 1
When u = 0 and v ≧ 1, the following equation is used instead of equation (D26).
Y≡E ^(v) U ′ ₁ U ′ ₂ ... U ′ _m−1 U ′ _m L ′ _m L ′ _m−1 ... L ′ ₂ L ′ ₁ AX
Here, the matrix B ′ is obtained by performing v times of column replacement on the matrix B. The matrix L ′ _i (i = 1 to m) is a basic lower triangular matrix of the lower triangular matrix L ′ obtained by LU decomposition of the matrix B ′. The matrix U ′ _i (i = 1 to m) is a basic upper triangular matrix of the upper triangular matrix U ′ obtained by LU decomposition of the matrix B ′.

The ordering unit reads out data representing the matrix B from the intermediate result storage unit, performs the ordering of the matrix B, generates a matrix B ′ in which the matrix B is replaced v times, and generates an intermediate result storage unit. And a matrix E ^{(v) in} which the same replacement as the column replacement for the matrix B is performed for the unit matrix E is generated and stored in the matrix storage unit.

Matrix storage unit, the basic upper triangular matrix _{U '1, U' 2,} ···, ' data representing the _m, basic lower triangular matrix _{L' U m, L 'm} -1, ···, L' ₁ Data representing and data representing the matrices A and E ^(v) are stored.

  In order to calculate the parity vector Y, the matrix product calculation unit executes (2m + 2) times of matrix multiplications included in the equation (D26) "in the order instructed by the matrix product calculation control unit.

In order to calculate the parity vector Y, the matrix product calculation control unit determines that the order of the matrices in which data is stored in the matrix storage unit is A, L ′ ₁ , L ′ ₂ ,..., L ′ _m , The matrix product calculation unit is set so that the matrix order is set so that U ′ _m , U ′ _m−1 ,..., U ′ ₁ , E ^(v) are in this order, and matrix multiplication is performed in this order. Control. The matrix product calculation control unit outputs a parity vector Y as a calculation result from the matrix product calculation unit after the (2m + 2) times of matrix multiplications included in the equation (D26) ″.

  The intermediate result storage unit stores data representing the intermediate result in the process (matrix decomposition process) for generating the matrix included in the expression (D26) ". Further, the intermediate result storage unit performs the parity vector Y generation process, Data representing an intermediate result of (2m + 2) times of matrix multiplication included in the expression (D26) "is stored.

(5-2) Modification 1 of the second embodiment
(5-2A) u ≧ 1 and v ≧ 1
In the first modification of the second embodiment, when u ≧ 1 and v ≧ 1, Expression (E23) is used.
Y = -E ^(v) U ' ₁ ^-1 U' ₂ ^-1 ... U ' _m-1 ^-1 U' _m ^-1 L ' _m ^-1 L' _m-1 ^-1 ... L ' ₂ ^-1 L ' ₁ ^-1 A ^(u) X (E23)

(5-2B) u ≧ 1 and v = 0
When u ≧ 1 and v = 0, the following equation is used instead of the equation (E23).
Y = -U ' ₁ ^-1 U' ₂ ^-1 ... U ' _m-1 ^-1 U' _m ^-1 L ' _m ^-1 L' _m-1 ^-1 ... L ' ₂ ^-1 L' ₁ ^-1 A ^(V) X (E23) '
Here, the matrix B ′ is obtained by performing u times of row replacement on the matrix B. The matrix L ′ _i ⁻¹ (i = ^{1 to} m) is an inverse matrix of the matrix L ′ _i . The matrix L ′ _i is a basic lower triangular matrix of the lower triangular matrix L ′ obtained by LU decomposition of the matrix B ′. The matrix U ′ _i ⁻¹ (i = ^{1 to} m) is an inverse matrix of the matrix U ′ _i . The matrix U ′ _i is a basic upper triangular matrix of the upper triangular matrix U ′ obtained by LU decomposition of the matrix B ′.

The ordering unit is the same as (5-1-B).
The matrix storage unit includes data representing the inverse matrix U ′ ₁ ⁻¹ , U ′ ₂ ⁻¹ ,..., U ′ _m ⁻¹ of the basic upper triangular matrix, the inverse matrix L ′ _m ^{−1 of} the basic lower triangular matrix, Data representing L ′ _m−1 ⁻¹ ,..., L ′ ₁ ⁻¹ and data representing the matrix A ^(u) are stored.

  In order to calculate the parity vector Y, the matrix product calculation unit executes (2m + 1) times of matrix multiplications included in the equation (E23) ′ in the order instructed by the matrix product calculation control unit.

In order to calculate the parity vector Y, the matrix product calculation control unit determines that the order of the matrices stored in the matrix storage unit is A ^(u) , L ′ ₁ ⁻¹ , L ′ ₂ ^−1,. ..L ' _m-1 ^-1 , L' _m ^-1 , U ' _m ^-1 , U' _m-1 ^-1 , ..., U ' ₂ ^-1 , U' ₁ ^-1 In this way, the order between the matrices is set, and the matrix product calculation unit is controlled to perform matrix multiplication in this order. Further, the matrix product calculation control unit, after the completion of (2m + 1) times of matrix multiplication included in the equation (E23) ′, the vector Y ′ (= U ′ ₁ ⁻¹ U ′ _{2) as a} calculation result from the matrix product calculation unit. ^-1 ... U ' _m-1 ^-1 U' _m ^-1 L ' _m ^-1 L' _m-1 ^-1 ... L ' ₂ ^-1 L' ₁ ^-1 A ^(u) ) To the output.

  The intermediate result storage unit stores data representing an intermediate result in a process (matrix decomposition process) for generating a matrix included in the equation (E23) ′. The intermediate result storage unit stores data representing the intermediate result of (2m + 1) times of matrix multiplication included in the equation (E23) ′ during the generation process of the parity vector Y.

(5-2C) u = 0 and v ≧ 1
When u = 0 and v ≧ 1, the following equation is used instead of the equation (E23).
^{Y = -E (v) U '} 1 -1 U' 2 -1 ··· U 'm-1 -1 U' m -1 L 'm -1 L' m-1 -1 ··· L '2 ^-1 L ' ₁ ^-1 AX (E23) "
Here, the matrix B ′ is obtained by performing v times of column replacement on the matrix B. The matrix L ′ _i ⁻¹ (i = ^{1 to} m) is an inverse matrix of the matrix L ′ _i . The matrix L ′ _i is a basic lower triangular matrix of the lower triangular matrix L ′ obtained by LU decomposition of the matrix B ′. The matrix U ′ _i ⁻¹ (i = ^{1 to} m) is an inverse matrix of the matrix U ′ _i . The matrix U ′ _i is a basic upper triangular matrix of the upper triangular matrix U ′ obtained by LU decomposition of the matrix B ′.

The ordering unit is the same as (5-2-2-B).
The matrix storage unit includes data representing the inverse matrix U ′ ₁ ⁻¹ , U ′ ₂ ⁻¹ ,..., U ′ _m ⁻¹ of the basic upper triangular matrix, the inverse matrix L ′ _m ^{−1 of} the basic lower triangular matrix, Data representing L ′ _m−1 ⁻¹ ,..., L ′ ₁ ⁻¹ and data representing the matrices A and E ^(v) are stored.

  In order to calculate the parity vector Y, the matrix product calculation unit executes (2m + 2) times of matrix multiplications included in the equation (E23) "in the order instructed by the matrix product calculation control unit.

In order to calculate the parity vector Y, the matrix product calculation control unit is arranged such that the order of the matrixes in which data is stored in the matrix storage unit is A, L ′ ₁ ⁻¹ , L ′ ₂ ⁻¹ ,. L ′ _m−1 ⁻¹ , L ′ _m ⁻¹ , U ′ _m ⁻¹ , U ′ _m−1 ⁻¹ ,..., U ′ ₂ ⁻¹ , U ′ ₁ ⁻¹ , E ^(v) The order between the matrices is set so as to satisfy the following, and the matrix product calculation unit is controlled to perform matrix multiplication in this order. Further, the matrix product calculation control unit, after the completion of (2m + 2) times of matrix multiplication included in the equation (E23) ", the vector Y ′ (= E ^(v) U ′ ₁ ^{− 1} U ' ₂ ^-1 ... U' _m-1 ^-1 U ' _m ^-1 L' _m ^-1 L ' _m-1 ^-1 ... L' ₂ ^-1 L ' ₁ ^-1 A) Output to the inverting unit.

  The intermediate result storage unit stores data representing an intermediate result in the process (matrix decomposition process) for generating the matrix included in the expression (E23) ". Further, the intermediate result storage unit performs the parity vector Y generation process, Data representing an intermediate result of (2m + 2) times of matrix multiplication included in the expression (E23) "is stored.

(5-3) Fourth Embodiment (5-3-A) u ≧ 1 and v ≧ 1
In the fourth embodiment, when u ≧ 1 and v ≧ 1, Expression (H26) is used, and the product of t matrices of E ^(v) and U ′ _{1 to} U ′ _t−1 is expressed as one matrix. When represented by C, formula (H28) was obtained.
Y≡E ^(v) U ′ ₁ U ′ ₂ ... U ′ _m−1 U ′ _m L ′ _m L ′ _m−1 ... L ′ ₂ L ′ ₁ A ^(u) X (H26 )
Y≡CU ' _t ... U' _m-1 U ' _m L' _m L ' _m-1 ... L' ₂ L ' ₁ A ^(u) X (H28)

(5-3-B) u ≧ 1 and v = 0
When u ≧ 1 and v = 0, the expression (H26) ′ is used instead of the expression (H26), and the product of t matrices of U ′ _{1 to} U ′ _t is represented by one matrix C Thus, the equation (H28) "is obtained.
Y≡U ′ ₁ U ′ ₂ ... U ′ _m−1 U ′ _m L ′ _m L ′ _m−1 ... L ′ ₂ L ′ ₁ A ^(u) X
_{Y≡CU 't + 1 ··· U'} m-1 U 'm L' m L 'm-1 ··· L' 2 L '1 A (u) X ··· (H28)'
Here, the matrix B ′ is obtained by performing u times of row replacement on the matrix B. The matrix L ′ _i (i = 1 to m) is a basic lower triangular matrix of the lower triangular matrix L ′ obtained by LU decomposition of the matrix B ′. The matrix U ′ _i (i = 1 to m) is a basic upper triangular matrix of the upper triangular matrix U ′ obtained by LU decomposition of the matrix B ′.

The ordering unit is the same as (5-1-B).
The matrix storage unit includes data representing basic upper triangular matrices U ′ _{t + 1} , U ′ _{t + 2} ,..., U ′ _m , basic lower triangular matrices L ′ _m , L ′ _m−1 ,. Data representing L ′ ₁ and data representing the matrices C and A ^(u) are stored.

  In order to calculate the parity vector Y, the matrix product calculation unit executes (2m−t + 2) matrix multiplications included in the equation (H28) ′ in the order instructed by the matrix product calculation control unit. The matrix product calculation unit executes the matrix multiplication instructed by the matrix multiplication number reduction unit when executing the process of reducing the number of matrix multiplications included in Expression (H26) ′.

When performing the process of reducing the number of matrix multiplications included in the expression (H26) ′, the matrix multiplication number reduction unit determines that the order of the matrices in which data is stored in the intermediate result storage unit is U ′ ₁ , U ′ ₂ ,..., U ′ _m−1 , U ′ _m , L ′ _m , L ′ _m−1 ,..., L ′ ₂ , L ′ ₁ , A ^(u) in order The matrix product calculation unit is controlled so as to perform matrix multiplication in this order.

In order to calculate the parity vector Y, the matrix product calculation control unit determines that the order of the matrices in which data is stored in the matrix storage unit is A ^(u) , L ′ ₁ , L ′ ₂ ,. ′ M ₋₁ , L ′ _m , U ′ _m , U ′ _m−1 ,..., U ′ _{t + 1} , C are set in order, and matrix multiplication is performed in this order. Control the matrix product calculator to do so. Further, the matrix product calculation control unit outputs the parity vector Y as a calculation result from the matrix product calculation unit after the (2m−t + 2) times of matrix multiplication included in the equation (H28) ′.

  The intermediate result storage unit stores data representing an intermediate result in a process (matrix decomposition process) for generating a matrix included in Expression (H28) ′. The intermediate result storage unit stores data representing the intermediate result of (2m−t + 2) times of matrix multiplication included in the expression (H28) ′ during the generation process of the parity vector Y.

(5-3-C) u = 0 and v ≧ 1
When u = 0 and v ≧ 1, the expression (H26) "is used instead of the expression (H26), and the product of t matrices of E ^(v) and U ' _{1 to} U' _t-1 is used. Is represented by one matrix C, the equation (H28) "is obtained.
Y≡E ^(v) U ' ₁ U' ₂ ... U ' _m-1 U' _m L ' _m L' _m-1 ... L ' ₂ L' ₁ AX (H26) "
Y≡CU ′ _t ... U ′ _m−1 U ′ _m L ′ _m L ′ _m−1 ... L ′ ₂ L ′ ₁ AX (H28) ”
Here, the matrix B ′ is obtained by performing v times of column replacement on the matrix B. The matrix L ′ _i (i = 1 to m) is a basic lower triangular matrix of the lower triangular matrix L ′ obtained by LU decomposition of the matrix B ′. The matrix U ′ _i (i = 1 to m) is a basic upper triangular matrix of the upper triangular matrix U ′ obtained by LU decomposition of the matrix B ′.

The ordering unit is the same as (5-1-C).
The matrix storage unit stores basic upper triangular matrices U ′ _t , U ′ _{t + 1} ,..., U ′ _m , basic lower triangular matrices L ′ _m , L ′ _m−1 ,. Data representing ₁ and data representing matrices C and A are stored.

  In order to calculate the parity vector Y, the matrix product calculation unit executes (2m−t + 3) times of matrix multiplications included in the formula (H28) ″ in the order instructed by the matrix product calculation control unit. When executing the process of reducing the number of matrix multiplications included in the formula (H26) ", the unit executes the matrix multiplication instructed by the matrix multiplication number reduction unit.

When performing the process of reducing the number of matrix multiplications included in the formula (H26) ", the matrix multiplication number reduction unit starts from the top of the matrix in which the data is stored in the intermediate result storage unit by E ^(v) , U ′ ₁ , U ′ ₂ ,..., U ′ _m−1 , U ′ _m , L ′ _m , L ′ _m−1 ,..., L ′ ₂ , L ′ ₁ , A The matrix product calculation unit is controlled so as to perform matrix multiplication in this order.

In order to calculate the parity vector Y, the matrix product calculation control unit determines that the order of the matrices in which data is stored in the matrix storage unit is A, L ′ ₁ , L ′ ₂ ,..., L ′ _{m− 1} , L ′ _m , U ′ _m , U ′ _m−1 ,..., U ′ _t , C are set in order, and the matrix product is set so that matrix multiplication is performed in this order. Control the calculator. In addition, the matrix product calculation control unit causes the matrix product calculation unit to output a parity vector Y as a calculation result after the (2m−t + 3) times of matrix multiplication included in the expression (H28) ″.

  The intermediate result storage unit stores data representing the intermediate result in the process (matrix decomposition process) for generating the matrix included in the formula (H28) ". The intermediate result storage unit Data representing an intermediate result of (2m−t + 3) times of matrix multiplication included in the expression (H28) ″ is stored.

(5-4) Modification 1 of the fourth embodiment
(5-4-A) u ≧ 1 and v ≧ 1
In the first modification of the fourth embodiment, when u ≧ 1 and v ≧ 1, the formula (I23) is used, and _{t of} E ^(v) , U ′ ₁ ^{−1 to} U ′ _t−1 ⁻¹ . Equation (I25) was obtained when the product of the matrices of
^{Y = -E (v) U '} 1 -1 U' 2 -1 ··· U 'm-1 -1 U' m -1 L 'm -1 L' m-1 -1 ··· L '2 ^-1 L ' ₁ ^-1 A ^(u) X (I23)
Y = −CU ′ _t ⁻¹ ... U ′ _m−1 ⁻¹ U ′ _m ⁻¹ L ′ _m ⁻¹ L ′ _m−1 ⁻¹ ... L ′ ₂ ⁻¹ L ′ ₁ ⁻¹ A ^{( u)} X (I25)

(5-4-B) u ≧ 1 and v = 0
When u ≧ 1 and v = 0, the equation (I23) ′ is used instead of the equation (I23), and the product of t matrices from U ′ ₁ ^{−1 to} U ′ _t ⁻¹ is expressed as one matrix C In this case, the formula (I25) ′ is obtained.
Y = -U ' ₁ ^-1 U' ₂ ^-1 ... U ' _m-1 ^-1 U' _m ^-1 L ' _m ^-1 L' _m-1 ^-1 ... L ' ₂ ^-1 L' ₁ ^-1 A ^(V) X (I23) '
Y = -CU ' _{t + 1} ^-1 ... U' _m-1 ^-1 U ' _m ^-1 L' _m ^-1 L ' _m-1 ^-1 ... L' ₂ ^-1 L ' ₁ ^-1 A ^(u) X (I25) '
Here, the matrix B ′ is obtained by performing u times of row replacement on the matrix B. The matrix L ′ _i ⁻¹ (i = ^{1 to} m) is an inverse matrix of the matrix L ′ _i . The matrix L ′ _i is a basic lower triangular matrix of the lower triangular matrix L ′ obtained by LU decomposition of the matrix B ′. The matrix U ′ _i ⁻¹ (i = ^{1 to} m) is an inverse matrix of the matrix U ′ _i . The matrix U ′ _i is a basic upper triangular matrix of the upper triangular matrix U ′ obtained by LU decomposition of the matrix B ′.

The ordering unit is the same as (5-1-B).
The matrix storage unit includes data representing the inverse matrices U ′ _{t + 1} ⁻¹ , U ′ _{t + 2} ⁻¹ ,..., U ′ _m ⁻¹ of the basic upper triangular matrix, and the inverse matrix L ′ of the basic lower triangular matrix Data representing _m ⁻¹ , L ′ _m−1 ⁻¹ ,..., L ′ ₁ ⁻¹ and data representing the matrices C and A ^(u) are stored.

  In order to calculate the parity vector Y, the matrix product calculation unit performs (2m−t + 2) times of matrix multiplications included in the equation (I25) ′ in the order instructed by the matrix product calculation control unit. The matrix product calculation unit executes the matrix multiplication instructed by the matrix multiplication number reduction unit when executing the process of reducing the number of matrix multiplications included in the formula (I23) ′.

When performing the process of reducing the number of matrix multiplications included in the formula (I23) ′, the matrix multiplication number reduction unit determines that the order of the matrices in which data is stored in the intermediate result storage unit is U ′ ₁ ⁻¹ from the beginning. ^{_{, U '2 -1, ···,}} U' m-1 -1, U 'm -1, L' m -1, L 'm-1 -1, ···, L' 2 -1, L The order between the matrices is set so as to be in the order of ′ ₁ ⁻¹ and A ^(u) , and the matrix product calculation unit is controlled to perform matrix multiplication in this order.

In order to calculate the parity vector Y, the matrix product calculation control unit determines that the order of the matrices in which data is stored in the matrix storage unit is A ^(u) , L ′ ₁ ⁻¹ , L ′ ₂ ^−1,. ..L ′ _m−1 ⁻¹ , L ′ _m ⁻¹ , U ′ _m ⁻¹ , U ′ _m−1 ⁻¹ ,..., U ′ _{t + 1} ⁻¹ , and C The order between the matrices is set, and the matrix product calculation unit is controlled to perform matrix multiplication in this order. Further, the matrix product calculation control unit, after the completion of (2m−t + 2) matrix multiplications included in the formula (I25) ′, the vector Y ′ (= CU ′ _{t + 1} ^{−) as the} calculation result from the matrix product calculation unit. ¹ ... U ' _m-1 ^-1 U' _m ^-1 L ' _m ^-1 L' _m-1 ^-1 ... L ' ₂ ^-1 L' ₁ ^-1 A ^(u) X) To the output.

  The intermediate result storage unit stores data representing an intermediate result in a process (matrix decomposition process) for generating a matrix included in the formula (I25) ′. The intermediate result storage unit stores data representing the intermediate result of (2m−t + 2) times of matrix multiplication included in the expression (I25) ′ during the generation process of the parity vector Y.

(5-4-C) u = 0 and v ≧ 1
Further, when u = 0 and v ≧ 1, the formula (I23) "is used instead of the formula (I23 ⁾ , and _t ⁽ E ^{) (v)} , U ' ₁ ^{-1 to} U' _t-1 ^-1 When the product of these matrices is represented by one matrix C, the equation (I25) "is obtained.
^{Y = -E (v) U '} 1 -1 U' 2 -1 ··· U 'm-1 -1 U' m -1 L 'm -1 L' m-1 -1 ··· L '2 ^-1 L ' ₁ ^-1 AX (I23) "
Y = -CU ' _t ^-1 ... U' _m-1 ^-1 U ' _m ^-1 L' _m ^-1 L ' _m-1 ^-1 ... L' ₂ ^-1 L ' ₁ ^-1 AX .. (I25) "
Here, the matrix B ′ is obtained by performing v times of column replacement on the matrix B. The matrix L ′ _i ⁻¹ (i = ^{1 to} m) is an inverse matrix of the matrix L ′ _i . The matrix L ′ _i is a basic lower triangular matrix of the lower triangular matrix L ′ obtained by LU decomposition of the matrix B ′. The matrix U ′ _i ⁻¹ (i = ^{1 to} m) is an inverse matrix of the matrix U ′ _i . The matrix U ′ _i is a basic upper triangular matrix of the upper triangular matrix U ′ obtained by LU decomposition of the matrix B ′.

The ordering unit is the same as (5-1-C).
The matrix storage unit includes data representing the inverse matrices U ′ _t ⁻¹ , U ′ _{t + 1} ⁻¹ ,..., U ′ _m ⁻¹ of the basic upper triangular matrix, and the inverse matrix L ′ _m ^{− of the} basic lower triangular matrix. ¹ , L ′ _m−1 ⁻¹ ,..., Data representing L ′ ₁ ⁻¹ and data representing the matrices C and A are stored.

  In order to calculate the parity vector Y, the matrix product calculation unit executes (2m−t + 3) times of matrix multiplications included in the formula (I25) ″ in the order instructed by the matrix product calculation control unit. When executing the process of reducing the number of matrix multiplications included in the formula (I23) ", the unit executes the matrix multiplication specified by the matrix multiplication number reduction unit.

When performing the process of reducing the number of matrix multiplications included in the formula (I23) ", the matrix multiplication number reduction unit starts from the top of the matrix in which the data is stored in the intermediate result storage unit by E ^(v) , U ′ ₁ ⁻¹ , U ′ ₂ ⁻¹ ,..., U ′ _m−1 ⁻¹ , U ′ _m ⁻¹ , L ′ _m ⁻¹ , L ′ _m−1 ^−1,. The order between the matrices is set so that ₂ ⁻¹ , L ′ ₁ ⁻¹ and A are in the order, and the matrix product calculation unit is controlled to perform matrix multiplication in this order.

In order to calculate the parity vector Y, the matrix product calculation control unit is arranged such that the order of the matrixes in which data is stored in the matrix storage unit is A, L ′ ₁ ⁻¹ , L ′ ₂ ⁻¹ ,. ^{_{L 'm-1 -1, L}} ' m -1, U 'm -1, U' m-1 -1, ···, U 't -1, the order between the matrix such that the order of C The matrix product calculation unit is controlled so that matrix multiplication is performed in this order. Further, the matrix product calculation control unit, after the completion of (2m−t + 3) times of matrix multiplication included in the formula (I25) ″, the vector Y ′ (= CU ′ _t ⁻¹ ...) From the matrix product calculation unit. U ′ _m−1 ⁻¹ U ′ _m ⁻¹ L ′ _m ⁻¹ L ′ _m−1 ⁻¹ ... L ′ ₂ ⁻¹ L ′ ₁ ⁻¹ AX) is output to the sign inversion unit.

  The intermediate result storage unit stores data representing the intermediate result in the process (matrix decomposition process) for generating the matrix included in the formula (I25) ". The intermediate result storage unit Data representing an intermediate result of (2m−t + 3) times of matrix multiplication included in the formula (I25) ″ is stored.

(6) Ordering method in which the number of non-zero elements of non-diagonal terms does not increase in the process of calculating the parity vector Y. For example, the matrix U ₃ and the matrix U ₄ are expressed by the equations (13) and (14), respectively. In this case, these products U ₄ × U ₃ are expressed by equation (15).

When the calculation result of the matrix U ₄ × U ₃ is observed, the fourth row is the same as the fourth row of U ₄ . The third row is the same as the third row of U ₃ when U [3,4] is zero. In the third row, when U [3, 4] is 1, the i-th column is U [3, i] + U [4, i]. In an operation modulo 2, when U [3, i] and U [4, i] are both 1 or both are 0, U [3, i] + U [4, i] The value of becomes zero.

This means that in order to reduce the number of non-zero elements in the non-diagonal terms of the matrix U ₄ × U ₃ , the matrix having the value “1” is adjacent to each other in each column of the upper triangular matrix U. This indicates that it is desirable to order B. In other words, in each column of the upper triangular matrix U, matrix multiplication is performed in order to calculate an expression for generating the parity vector Y by ordering the matrix B so that elements having a value of “1” are adjacent to each other. However, the number of non-zero elements in the off-diagonal terms can be prevented from increasing. In addition, by performing such ordering, the number of non-zero elements of non-diagonal terms does not increase even if a plurality of matrices are combined into one, so that the number of matrix multiplications can be effectively reduced by matrix multiplication reduction processing. can do.

Further, when the matrix L ₃ and the matrix L ₄ are represented by the equations (16) and (17), respectively, the product L ₄ × L ₃ is represented by the equation (18).

Observing the calculation result of the matrix L ₄ × L ₃ , the fourth column is the same as the fourth column of L ₄ . The third column is the same as the third column of L ₃ when L [4,3] is zero. In the third column, when L [4,3] is 1, the i-th row is L [i, 3] + L [i, 4]. In an operation modulo 2, if both L [i, 3] and L [i, 4] are 1, or both are 0, L [i, 3] + L [i, 4] The value of becomes zero.

This means that in order to reduce the number of non-zero elements in the non-diagonal terms of the matrix L ₄ × L ₃ , the matrix B Indicates that it is desirable to order In other words, in each row of the lower triangular matrix L, matrix multiplication is performed to calculate an expression for generating the parity vector Y by ordering the matrix B so that elements having a value of “1” are adjacent to each other. However, the number of non-zero elements in the off-diagonal terms can be prevented from increasing. In addition, by performing such ordering, the number of non-zero elements of non-diagonal terms does not increase even if a plurality of matrices are combined into one, so that the number of matrix multiplications can be effectively reduced by matrix multiplication reduction processing. can do.

(7) Encoding program The encoder described in the embodiment of the present invention is not limited to that realized by a dedicated hardware device. The encoding program may be externally installed in the memory, and the function of the encoder may be realized by the computer reading the encoding program from the memory and executing it. In this case, the encoding program for executing the functions of the first embodiment includes the steps of the flowcharts of FIGS. 3 and 4 and has the same effect as that of the first embodiment. The encoding program for executing the functions of the first modification of the first embodiment includes the steps of the flowcharts of FIGS. 6 and 7 and has the same effect as the modification of the first embodiment. The encoding program for executing the functions of the second embodiment includes the steps of the flowcharts of FIGS. 9 and 10 and has the same effects as those of the second embodiment. The encoding program for executing the function of the first modification of the second embodiment includes the steps of the flowcharts of FIGS. 12 and 13 and has the same effect as that of the first modification of the second embodiment. The encoding program for executing the functions of the third embodiment includes the steps of the flowcharts of FIGS. 15 and 16 and has the same effects as those of the third embodiment. The encoding program that executes the function of the first modification of the third embodiment includes the steps of the flowcharts of FIGS. 18 and 19 and has the same effect as that of the first modification of the third embodiment. The encoding program for executing the functions of the fourth embodiment includes the steps of the flowcharts of FIGS. 21 and 22 and has the same effects as those of the fourth embodiment. The encoding program for executing the functions of the first modification of the fourth embodiment includes the steps of the flowcharts of FIGS. 24 and 25, and has the same effect as that of the first modification of the fourth embodiment. The encoding program for executing the functions of the fifth embodiment includes the steps of the flowcharts of FIGS. 3 and 27, and has the same effect as that of the fifth embodiment.

(8) Error Correction Code In the embodiment of the present invention, LDPC has been described as an example of an error correction code. However, the present invention is not limited to this, and application to a Viterbi or turbo code is also possible.

(9) (t + 1) <m, t <m
In the third embodiment of the present invention and its modification example 1, (t + 1) <m is set for convenience of explanation, but the present invention is not limited to this. Similarly, in the fourth embodiment of the present invention and its modification example 1, t <m is set for convenience of explanation, but the present invention is not limited to this.

  The embodiment disclosed this time should be considered as illustrative in all points and not restrictive. The scope of the present invention is defined by the terms of the claims, rather than the description above, and is intended to include any modifications within the scope and meaning equivalent to the terms of the claims.

It is a figure which shows an example of a structure of the communication system of embodiment of this invention. It is a figure showing the structure of the encoder of 1st Embodiment. It is a flowchart of the matrix decomposition process of 1st Embodiment. It is a flowchart of the encoding vector production | generation process of 1st Embodiment. It is a figure showing the structure of the encoder of the modification 1 of 1st Embodiment. It is a flowchart of the matrix decomposition process of the modification 1 of 1st Embodiment. It is a flowchart of the encoding vector production | generation process of the modification 1 of 1st Embodiment. It is a figure showing the structure of the encoder of 2nd Embodiment. It is a flowchart of the matrix decomposition process of 2nd Embodiment. It is a flowchart of the encoding vector production | generation process of 2nd Embodiment. It is a figure showing the structure of the encoder of the modification 1 of 2nd Embodiment. It is a flowchart of the matrix decomposition process of the modification 1 of 2nd Embodiment. It is a flowchart of the encoding vector production | generation process of the modification 1 of 2nd Embodiment. It is a figure showing the structure of the encoder of 3rd Embodiment. It is a flowchart of the matrix decomposition process of 3rd Embodiment. It is a flowchart of the encoding vector production | generation process of 3rd Embodiment. It is a figure showing the structure of the encoder of the modification 1 of 3rd Embodiment. It is a flowchart of the matrix decomposition process of the modification 1 of 3rd Embodiment. It is a flowchart of the encoding vector production | generation process of the modification 1 of 3rd Embodiment. It is a figure showing the structure of the encoder of 4th Embodiment. It is a flowchart of the matrix decomposition process of 4th Embodiment. It is a flowchart of the encoding vector production | generation process of 4th Embodiment. It is a figure showing the structure of the encoder of the modification 1 of 4th Embodiment. It is a flowchart of the matrix decomposition process of the modification 1 of 4th Embodiment. It is a flowchart of the encoding vector production | generation process of the modification 1 of 4th Embodiment. It is a figure showing the structure of the encoder of 5th Embodiment. It is a flowchart of the encoding vector production | generation process of 5th Embodiment.

Explanation of symbols

  DESCRIPTION OF SYMBOLS 1 Transmitting device, 2, 2a-2i encoder, 3 modulator, 4 receiving device, 5 error correction decoder, 6 demodulator, 10, 30 Check matrix decomposing unit, 11, 21, 31, 41, 51, 61, 71, 81, 91 Matrix product calculation unit, 97 First matrix product calculation unit, 98 Second matrix product calculation unit, 12, 22, 32, 42, 52, 62, 72, 82, 92 Intermediate result storage Part 13, 17, 27, 33, 37, 47, 57, 67, 77, 87, 94, 96 adder, 14, 34 LU decomposition part, 15, 19, 29, 35, 39, 49, 59, 69 79, 89, 93, 95 multiplier, 16, 36 basic matrix decomposition unit, 18, 28, 38, 48, 58, 68, 78, 88 matrix storage unit, 20 array unit, 23, 43 inverse matrix calculation unit, 40 ordering unit, 75,85 matrix multiplication number reduction unit , 99 sign inversion unit, 121, 122, 123, 124, 125, 126, 127, 128, 129 matrix product calculation control unit.

Claims (24)

An encoding device that generates a parity vector Y having m elements by performing an operation modulo 2 from an information vector X having k elements in accordance with a parity generation expression represented by Expression (A1),
Y≡U ₁ U ₂ ... U _m-1 U _m L _m L _m-1 ... L ₂ L ₁ AX (A1)
However, (X, Y) ^T represents transposition of the vector (X, Y), the parity check matrix H is an m-row n-column matrix satisfying H (X, Y) ^T ≡0, and the matrix A is , The first column to the k-th column of the parity check matrix H, the matrix B includes the (k + 1) -th column to the n-th column of the parity check matrix H, and n = k + m,
The matrix L _i (i = 1 to m) is a basic lower triangular matrix of the lower triangular matrix L obtained by LU decomposition of the matrix B, and the i-th column matches the i-th column of the lower triangular matrix L. , The other columns are matrices with off-diagonal terms 0 and diagonal terms 1.
The matrix U _i (i = 1 to m) is a basic upper triangular matrix of the upper triangular matrix U obtained by LU decomposition of the matrix B, and the i-th row matches the i-th row of the upper triangular matrix U. , The other rows are matrices with off-diagonal terms of 0 and diagonal terms of 1,
The encoding device includes:
A matrix storage unit for storing data representing a plurality of matrices included in the parity generation equation;
An intermediate result storage unit for storing data representing intermediate results of a plurality of matrix multiplications included in the parity generation equation;
In order to calculate the parity vector Y, a matrix product calculation unit that performs a plurality of matrix multiplications included in the parity generation equation,
The matrix product calculation unit performs each matrix multiplication included in the parity generation formula using the information vector X, data representing a matrix in the matrix storage unit, or data representing an intermediate result in the intermediate result storage unit Until the matrix multiplication of the plurality of times included in the parity generation formula ends, data representing the calculation result is stored in the intermediate result storage unit as data representing the intermediate result, and the parity generation formula When the plurality of matrix multiplications included in is completed, a vector obtained after the completion of the plurality of matrix multiplications included in the parity generation equation is output as a parity vector Y. The encoding device is an encoding device that generates a matrix included in the parity generation equation before generating the parity vector Y,
The intermediate result storage unit further stores data representing an intermediate result of processing for generating a matrix included in the parity generation formula,
The encoding device further includes:
A parity check matrix H that is decomposed into the matrix A and the matrix B, and a parity check matrix decomposition unit that stores data representing the matrix A in the matrix storage unit;
LU which decomposes the matrix B into a product of the lower triangular matrix L and the upper triangular matrix U and stores the data representing the lower triangular matrix L and the data representing the upper triangular matrix U in the intermediate result storage unit A decomposition unit;
Data representing the upper triangular matrix U and the lower triangular matrix L is read from the intermediate result storage unit, and the upper triangular matrix U is converted into a product of m basic upper triangular matrices U _i (i = 1 to m). The lower triangular matrix L is decomposed into a product of m basic lower triangular matrices L _i (i = 1 to m), and the decomposed matrix U _i (i = 1 to m) and the matrix L _i are decomposed. The encoding apparatus according to claim 1, further comprising: a basic matrix decomposition unit that stores data representing (i = 1 to m) in the matrix storage unit. An encoding device that generates a parity vector Y having m elements by performing a real number operation from an information vector X having k elements in accordance with a parity generation expression represented by Expression (A2),
Y = −U ₁ ⁻¹ U ₂ ⁻¹ ... U _m-1 ⁻¹ U _m ⁻¹ L _m ⁻¹ L _m−1 ⁻¹ ... L ₂ ⁻¹ L ₁ ⁻¹ AX. A2)
However, (X, Y) ^T represents transposition of the vector (X, Y), the parity check matrix H is an m-by-n matrix satisfying H (X, Y) ^T = 0, and the matrix A is , The first column to the k-th column of the parity check matrix H, the matrix B includes the (k + 1) -th column to the n-th column of the parity check matrix H, and n = k + m,
Matrix L _i ^-1 (i = 1 to m) is the inverse matrix of the matrix L _i, the matrix L _i is a basic lower triangular matrix of the lower triangular matrix L obtained by the LU decomposition of the matrix B, the i-th row Eyes coincide with the i-th column of the lower triangular matrix L, and the other columns are matrices whose off-diagonal terms are 0 and diagonal terms are 1,
Matrix U _i ^-1 (i = 1 to m) is the inverse matrix of the matrix U _i, the matrix U _i is the matrix B a fundamental upper triangular matrix of the upper triangular matrix U obtained by the LU decomposition, the i The row matches the i-th row of the upper triangular matrix U, and the other rows are matrices in which the off-diagonal term is 0 and the diagonal term is 1.
The encoding device includes:
A matrix storage unit for storing data representing a plurality of matrices included in the parity generation equation;
An intermediate result storage unit for storing data representing intermediate results of a plurality of matrix multiplications included in the parity generation equation;
In order to calculate the parity vector Y, a matrix product calculator that performs a plurality of matrix multiplications included in the parity generation equation;
A code inversion unit that inverts the sign of the vector output from the matrix product calculation unit and outputs it as a parity vector Y,
The matrix product calculation unit performs each matrix multiplication included in the parity generation formula using the information vector X, data representing a matrix in the matrix storage unit, or data representing an intermediate result in the intermediate result storage unit Until the matrix multiplication of the plurality of times included in the parity generation formula ends, data representing the calculation result is stored in the intermediate result storage unit as data representing the intermediate result, and the parity generation formula When the plurality of matrix multiplications included in is completed, a vector obtained after the completion of the plurality of matrix multiplications included in the parity generation equation is output to the code inverting unit. The encoding device is an encoding device that generates a matrix included in the parity generation equation before generating the parity vector Y,
The intermediate result storage unit further stores data representing an intermediate result of processing for generating a matrix included in the parity generation formula,
The encoding device further includes:
A parity check matrix H that is decomposed into the matrix A and the matrix B, and a parity check matrix decomposition unit that stores data representing the matrix A in the matrix storage unit;
LU which decomposes the matrix B into a product of the lower triangular matrix L and the upper triangular matrix U and stores the data representing the lower triangular matrix L and the data representing the upper triangular matrix U in the intermediate result storage unit A decomposition unit;
Data representing the upper triangular matrix U and the lower triangular matrix L is read from the intermediate result storage unit, and the upper triangular matrix U is converted into a product of m basic upper triangular matrices U _i (i = 1 to m). The lower triangular matrix L is decomposed into a product of m basic lower triangular matrices L _i (i = 1 to m), and the decomposed matrix U _i (i = 1 to m) and the matrix L _i are decomposed. A basic matrix decomposition unit that stores data representing (i = 1 to m) in the intermediate result storage unit;
Data representing the matrix U _i (i = 1 to m) and the matrix L _i (i = 1 to m) is read from the intermediate result storage unit, and the matrix U _i (i = 1 to m) and the matrix By inverting the sign of the element of the off-diagonal term of L _i (i = 1 to m), the inverse matrix U _i ⁻¹ (i = ^{1 to} m) and the inverse matrix L _i ⁻¹ (i = ^{1 to} m) And an inverse matrix calculator that stores data representing the inverse matrix U _i ⁻¹ (i = ^{1 to} m) and the inverse matrix L _i ⁻¹ (i = ^{1 to} m) in the matrix storage unit; The encoding device according to claim 3, comprising: From the information vector X with k elements, equation (A3-1) is obtained when u ≧ 1 and v ≧ 1, equation (A3-2) is obtained when u ≧ 1 and v = 0, and when u = 0 and v ≧ 1. An encoding device that generates a parity vector Y having m elements by performing an operation modulo 2 according to a parity generation expression represented by Expression (A3-3),
Y≡E ^(v) U ′ ₁ U ′ ₂ ... U ′ _m−1 U ′ _m L ′ _m L ′ _m−1 ... L ′ ₂ L ′ ₁ A ^(u) X (A3 -1)
Y≡U ′ ₁ U ′ ₂ ... U ′ _m−1 U ′ _m L ′ _m L ′ _m−1 ... L ′ ₂ L ′ ₁ A ^(u) X (A3-2)
Y≡E ^(v) U ′ ₁ U ′ ₂ ... U ′ _m−1 U ′ _m L ′ _m L ′ _m−1 ... L ′ ₂ L ′ ₁ AX (A3-3)
However, (X, Y) ^T represents transposition of the vector (X, Y), the parity check matrix H is an m-row n-column matrix satisfying H (X, Y) ^T ≡0, and the matrix A is , The first column to the k-th column of the parity check matrix H, the matrix B includes the (k + 1) -th column to the n-th column of the parity check matrix H, and n = k + m,
When u ≧ 1 and v ≧ 1, the matrix B ′ is obtained by performing u times of row replacement and v times of column replacement for the matrix B, and the matrix A ^(u) Is the same permutation as the u-th row permutation for the matrix B with respect to the matrix A, and the matrix E ^(v) is v times for the matrix B with respect to the unit matrix E. It is the same replacement as the column replacement,
When u ≧ 1 and v = 0, the matrix B ′ is obtained by performing u times of row replacement with respect to the matrix B, and the matrix A ^(u) is obtained with respect to the matrix A. , And the same replacement as the u-time replacement of the matrix B,
When u = 0 and v ≧ 1, the matrix B ′ is obtained by performing v times of column replacement with respect to the matrix B, and the matrix E ^(v) is obtained with respect to the unit matrix E. , The same permutation as the column permutation of the matrix B is performed,
The matrix L ′ _i (i = 1 to m) is a basic lower triangular matrix of the lower triangular matrix L ′ obtained by LU decomposition of the matrix B ′, and the i-th column is the i-th column of the lower triangular matrix L ′. The other column is a matrix whose off-diagonal term is 0 and whose diagonal term is 1,
The matrix U ′ _i (i = 1 to m) is a basic upper triangular matrix of the upper triangular matrix U ′ obtained by LU decomposition of the matrix B ′, and the i-th row is the i-th row of the upper triangular matrix U ′. The other row is a matrix whose off-diagonal terms are 0 and diagonal terms are 1,
The encoding device includes:
A matrix storage unit for storing data representing a plurality of matrices included in the parity generation equation;
An intermediate result storage unit for storing data representing intermediate results of a plurality of matrix multiplications included in the parity generation equation;
In order to calculate the parity vector Y, a matrix product calculation unit that performs a plurality of matrix multiplications included in the parity generation equation,
The matrix product calculation unit performs each matrix multiplication included in the parity generation formula using the information vector X, data representing a matrix in the matrix storage unit, or data representing an intermediate result in the intermediate result storage unit Until the matrix multiplication of the plurality of times included in the parity generation formula ends, data representing the calculation result is stored in the intermediate result storage unit as data representing the intermediate result, and the parity generation formula When the plurality of matrix multiplications included in is completed, a vector obtained after the completion of the plurality of matrix multiplications included in the parity generation equation is output as a parity vector Y. The encoding device is an encoding device that generates a matrix included in the parity generation equation before generating the parity vector Y,
The intermediate result storage unit further stores data representing an intermediate result of processing for generating a matrix included in the parity generation formula,
The encoding device further includes:
The parity check matrix H is decomposed into the matrix A and the matrix B, and when u = 0 and v ≧ 1, a check matrix decomposition unit that stores data representing the matrix A in the matrix storage unit;
When u ≧ 1 and v ≧ 1, the matrix B is ordered, and the matrix B ′ is generated by performing u times of row replacement and v times of column replacement for the matrix B, and the matrix B ′ is generated. The matrix A ^{(u) in} which the same replacement as the u-time replacement of the matrix B with respect to the matrix B is generated for A and stored in the matrix storage unit, and the matrix B for the unit matrix E with respect to the matrix B The matrix E ^(v) subjected to the same replacement as the column replacement of v times is generated and stored in the matrix storage unit. When u ≧ 1 and v = 0, the matrix B is ordered, The matrix B ′ in which u rows have been replaced with respect to the matrix B is generated, and the matrix A ^(u with the same replacement as u rows with respect to the matrix B is performed with respect to the matrix A. ⁾ generated and stored in the matrix storage unit, When = 0 and v ≧ 1, the matrix B is ordered to generate the matrix B ′ in which the columns B are exchanged v times, and the matrix B is converted to the unit matrix E. An ordering unit that generates the matrix E ^(v) that has been replaced in the same manner as the v-time column replacement for and stores the matrix E ^(v) in the matrix storage unit;
The matrix B ′ is LU-decomposed into a product of the lower triangular matrix L ′ and the upper triangular matrix U ′, and data representing the lower triangular matrix L ′ and data representing the upper triangular matrix U ′ are stored in the intermediate result. LU decomposition unit to be stored in the unit,
Data representing the upper triangular matrix U ′ and the lower triangular matrix L ′ is read from the intermediate result storage unit, and the upper triangular matrix U ′ is converted into the m basic upper triangular matrices U ′ _i (i = 1 to m). ), The lower triangular matrix L ′ is decomposed into products of m basic lower triangular matrices L ′ _i (i = 1 to m), and the decomposed matrix U ′ _i (i = 1). ~m) and matrix 'L _{i (i} = data representing the 1 to m) and a base matrix decomposition unit to be stored in the matrix storage unit, the encoding apparatus according to claim 5, wherein. From the information vector X with k elements, the equation (A4-1) is obtained when u ≧ 1 and v ≧ 1, the equation (A4-2) when u ≧ 1 and v = 0, and the equation (A4-2) when u = 0 and v ≧ 1. An encoding device that generates a parity vector Y having m elements by performing an operation modulo 2 according to a parity generation expression represented by Expression (A4-3),
^{Y = -E (v) U '} 1 -1 U' 2 -1 ··· U 'm-1 -1 U' m -1 L 'm -1 L' m-1 -1 ··· L '2 ^-1 L ' ₁ ^-1 A ^(u) X (A4-1)
Y = -U ' ₁ ^-1 U' ₂ ^-1 ... U ' _m-1 ^-1 U' _m ^-1 L ' _m ^-1 L' _m-1 ^-1 ... L ' ₂ ^-1 L' ₁ ^-1 A ^(V) X (A4-2)
^{Y = -E (v) U '} 1 -1 U' 2 -1 ··· U 'm-1 -1 U' m -1 L 'm -1 L' m-1 -1 ··· L '2 ^-1 L ' ₁ ^-1 AX (A4-3)
However, the parity check matrix H is a matrix of m rows and n columns satisfying H (X, Y) ^T = 0, and the matrix A is composed of the first to kth columns of the parity check matrix H. B consists of the (k + 1) th column to the nth column of the parity check matrix H, and n = k + m,
When u ≧ 1 and v ≧ 1, the matrix B ′ is obtained by performing u times of row replacement and v times of column replacement for the matrix B, and the matrix A ^(u) Is the same permutation as the u-th row permutation for the matrix B with respect to the matrix A, and the matrix E ^(v) is v times for the matrix B with respect to the unit matrix E. It is the same replacement as the column replacement,
When u ≧ 1 and v = 0, the matrix B ′ is obtained by performing u times of row replacement with respect to the matrix B, and the matrix A ^(u) is obtained with respect to the matrix A. , And the same replacement as the u-time replacement of the matrix B,
When u = 0 and v ≧ 1, the matrix B ′ is obtained by performing v times of column replacement with respect to the matrix B, and the matrix E ^(v) is obtained with respect to the unit matrix E. , The same permutation as the column permutation of the matrix B is performed,
Matrix ^{_{L 'i -1 (i = 1~m}} ) is a matrix L' is the inverse matrix of _i, the matrix L _'i, said matrix B' of the basic lower triangular matrix of LU decomposition lower triangular matrix has L ' The i-th column matches the i-th column of the lower triangular matrix L ′, and the other columns are matrices whose off-diagonal terms are 0 and whose diagonal terms are 1.
Matrix ^{_{U 'i -1 (i = 1~m}} ) is a matrix U' is the inverse matrix of _i, the matrix U _'i, said matrix B' of the basic upper triangular matrix of the upper triangular matrix U 'was LU decomposition The i-th row coincides with the i-th row of the upper triangular matrix U ′, and the other rows are matrices in which the off-diagonal term is 0 and the diagonal term is 1.
The encoding device includes:
A matrix storage unit for storing data representing a plurality of matrices included in the parity generation equation;
An intermediate result storage unit for storing data representing intermediate results of a plurality of matrix multiplications included in the parity generation equation;
In order to calculate the parity vector Y, a matrix product calculator that performs a plurality of matrix multiplications included in the parity generation equation;
A code inversion unit that inverts the sign of the vector output from the matrix product calculation unit and outputs it as a parity vector Y,
The matrix product calculation unit performs each matrix multiplication included in the parity generation formula using the information vector X, data representing a matrix in the matrix storage unit, or data representing an intermediate result in the intermediate result storage unit Until the matrix multiplication of the plurality of times included in the parity generation formula ends, data representing the calculation result is stored in the intermediate result storage unit as data representing the intermediate result, and the parity generation formula When the plurality of matrix multiplications included in is completed, a vector obtained after the completion of the plurality of matrix multiplications included in the parity generation equation is output to the code inverting unit. The encoding device is an encoding device that generates a matrix included in the parity generation equation before generating the parity vector Y,
The intermediate result storage unit further stores data representing an intermediate result of processing for generating a matrix included in the parity generation formula,
The encoding device further includes:
The parity check matrix H is decomposed into the matrix A and the matrix B, and when u = 0 and v ≧ 1, a check matrix decomposition unit that stores data representing the matrix A in the matrix storage unit;
When u ≧ 1 and v ≧ 1, the matrix B is ordered, and the matrix B ′ is generated by performing u times of row replacement and v times of column replacement for the matrix B, and the matrix B ′ is generated. The matrix A ^{(u) in} which the same replacement as the u-time replacement of the matrix B with respect to the matrix B is generated for A and stored in the matrix storage unit, and the matrix B for the unit matrix E with respect to the matrix B The matrix E ^(v) subjected to the same replacement as the column replacement of v times is generated and stored in the matrix storage unit. When u ≧ 1 and v = 0, the matrix B is ordered, The matrix B ′ in which u rows have been replaced with respect to the matrix B is generated, and the matrix A ^(u with the same replacement as u rows with respect to the matrix B is performed with respect to the matrix A. ⁾ generated and stored in the matrix storage unit, When = 0 and v ≧ 1, the matrix B is ordered to generate the matrix B ′ in which the columns B are exchanged v times, and the matrix B is converted to the unit matrix E. An ordering unit that generates the matrix E ^(v) that has been replaced in the same manner as the v-time column replacement for and stores the matrix E ^(v) in the matrix storage unit;
The matrix B ′ is LU-decomposed into a product of the lower triangular matrix L ′ and the upper triangular matrix U ′, and data representing the lower triangular matrix L ′ and data representing the upper triangular matrix U ′ are stored in the intermediate result. LU decomposition unit to be stored in the unit,
Data representing the upper triangular matrix U ′ and the lower triangular matrix L ′ is read from the intermediate result storage unit, and the upper triangular matrix U ′ is converted into the m basic upper triangular matrices U ′ _i (i = 1 to m). ), The lower triangular matrix L ′ is decomposed into products of m basic lower triangular matrices L ′ _i (i = 1 to m), and the decomposed matrix U ′ _i (i = 1). ˜m) and matrix ′ L _i (i = 1 to m) are stored in the intermediate result storage unit, a basic matrix decomposition unit;
The intermediate results from the storage unit reads the data representing the matrix U _{'i (i} = 1~m) and the matrix _{L' i (i = 1~m)} , the matrix U _{'i (i} = 1~m) And the inverse matrix U ′ _i ⁻¹ (i = ^{1 to} m) and the inverse matrix L ′ _i ⁻¹ by inverting the signs of the elements of the off-diagonal terms of the matrix L ′ _i (i = ^{1 to} m). (I = 1 to m) is calculated, and the matrix storage unit stores data representing the inverse matrix U ′ _i ⁻¹ (i = ^{1 to} m) and the inverse matrix L ′ _i ⁻¹ (i = ^{1 to} m). The encoding apparatus according to claim 7, further comprising: an inverse matrix calculation unit that stores the matrix. From the information vector X with the number k of elements, a parity generation expression in which at least one set of two or more adjacent matrices included in the expression (A5) is replaced with a matrix C that is a multiplication result of the two or more matrices. Accordingly, an encoding device that generates a parity vector Y having m elements by performing an operation modulo 2;
Y≡U ₁ U ₂ ... U _m-1 U _m L _m L _m-1 ... L ₂ L ₁ AX (A5)
However, (X, Y) ^T represents transposition of the vector (X, Y), the parity check matrix H is an m-row n-column matrix satisfying H (X, Y) ^T ≡0, and the matrix A is , The first column to the k-th column of the parity check matrix H, the matrix B includes the (k + 1) -th column to the n-th column of the parity check matrix H, and n = k + m,
The matrix L _i (i = 1 to m) is a basic lower triangular matrix of the lower triangular matrix L obtained by LU decomposition of the matrix B, and the i-th column matches the i-th column of the lower triangular matrix L. , The other columns are matrices with off-diagonal terms 0 and diagonal terms 1.
The matrix U _i (i = 1 to m) is a basic upper triangular matrix of the upper triangular matrix U obtained by LU decomposition of the matrix B, and the i-th row matches the i-th row of the upper triangular matrix U. , The other rows are matrices with off-diagonal terms of 0 and diagonal terms of 1,
The encoding device includes:
A matrix storage unit for storing data representing a plurality of matrices included in the parity generation equation;
An intermediate result storage unit for storing data representing intermediate results of a plurality of matrix multiplications included in the parity generation equation;
In order to calculate the parity vector Y, a matrix product calculation unit that performs a plurality of matrix multiplications included in the parity generation equation,
The matrix product calculation unit performs each matrix multiplication included in the parity generation formula using the information vector X, data representing a matrix in the matrix storage unit, or data representing an intermediate result in the intermediate result storage unit Until the matrix multiplication of the plurality of times included in the parity generation formula ends, data representing the calculation result is stored in the intermediate result storage unit as data representing the intermediate result, and the parity generation formula When the plurality of matrix multiplications included in is completed, a vector obtained after the completion of the plurality of matrix multiplications included in the parity generation equation is output as a parity vector Y.   The sum of the number of non-zero elements in the non-diagonal terms of all matrices included in the parity generation formula is equal to or less than the sum of the number of non-zero elements in the non-diagonal terms of all matrices included in the formula (A5). The encoding device according to claim 9. The encoding device is an encoding device that generates a matrix included in the parity generation equation before generating the parity vector Y,
The intermediate result storage unit further stores data representing an intermediate result of processing for generating a matrix included in the parity generation formula,
The matrix product calculation unit further performs multiplication of two or more adjacent matrices included in the formula (A5), and outputs data representing the matrix C as a multiplication result,
The encoding device further includes:
A parity check matrix H that is decomposed into the matrix A and the matrix B, and a parity check matrix decomposition unit that stores data representing the matrix A in the matrix storage unit;
LU which decomposes the matrix B into a product of the lower triangular matrix L and the upper triangular matrix U and stores the data representing the lower triangular matrix L and the data representing the upper triangular matrix U in the intermediate result storage unit A decomposition unit;
Data representing the upper triangular matrix U and the lower triangular matrix L is read from the intermediate result storage unit, and the upper triangular matrix U is converted into a product of m basic upper triangular matrices U _i (i = 1 to m). The lower triangular matrix L is decomposed into a product of m basic lower triangular matrices L _i (i = 1 to m), and the decomposed matrix U _i (i = 1 to m) and the matrix L _i are decomposed. A basic matrix decomposition unit that stores data representing (i = 1 to m) in the intermediate result storage unit;
The sum of the number of non-zero elements of the matrix C, which is the multiplication result, and the number of non-zero elements in the off-diagonal terms of the matrix excluding the two or more adjacent matrices among the matrices included in the equation (A5), Of the matrices included in the equation (A5) stored in the intermediate result storage unit when the number of non-zero elements of non-diagonal terms of all the matrices included in the equation (A5) is equal to or less than the sum 11. The encoding device according to claim 10, further comprising: a matrix multiplication number reduction unit that stores data representing a matrix excluding the two or more adjacent matrices and data representing the matrix C in the matrix storage unit. . From the information vector X having the number of elements k, a parity generation equation in which at least one set of two or more adjacent matrices included in the equation (A6) is replaced with a matrix C that is a multiplication result of the two or more matrices is used. Accordingly, an encoding device that generates a parity vector Y having m elements by performing a real number operation,
Y = −U ₁ ⁻¹ U ₂ ⁻¹ ... U _m-1 ⁻¹ U _m ⁻¹ L _m ⁻¹ L _m−1 ⁻¹ ... L ₂ ⁻¹ L ₁ ⁻¹ AX. A6)
However, (X, Y) ^T represents transposition of the vector (X, Y), the parity check matrix H is an m-by-n matrix satisfying H (X, Y) ^T = 0, and the matrix A is , The first column to the k-th column of the parity check matrix H, the matrix B includes the (k + 1) -th column to the n-th column of the parity check matrix H, and n = k + m,
Matrix L _i ^-1 (i = 1 to m) is the inverse matrix of the matrix L _i, the matrix L _i is a basic lower triangular matrix of the lower triangular matrix L obtained by the LU decomposition of the matrix B, the i-th row Eyes coincide with the i-th column of the lower triangular matrix L, and the other columns are matrices whose off-diagonal terms are 0 and diagonal terms are 1,
Matrix U _i ^-1 (i = 1 to m) is the inverse matrix of the matrix U _i, the matrix U _i is the matrix B a fundamental upper triangular matrix of the upper triangular matrix U obtained by the LU decomposition, the i The row matches the i-th row of the upper triangular matrix U, and the other rows are matrices in which the off-diagonal term is 0 and the diagonal term is 1.
The encoding device includes:
A matrix storage unit for storing data representing a plurality of matrices included in the parity generation equation;
An intermediate result storage unit for storing data representing intermediate results of a plurality of matrix multiplications included in the parity generation equation;
In order to calculate the parity vector Y, a matrix product calculator that performs a plurality of matrix multiplications included in the parity generation equation;
A code inversion unit that inverts the sign of the vector output from the matrix product calculation unit and outputs it as a parity vector Y,
The matrix product calculation unit performs each matrix multiplication included in the parity generation formula using the information vector X, data representing a matrix in the matrix storage unit, or data representing an intermediate result in the intermediate result storage unit Until the matrix multiplication of the plurality of times included in the parity generation formula ends, data representing the calculation result is stored in the intermediate result storage unit as data representing the intermediate result, and the parity generation formula When the plurality of matrix multiplications included in is completed, a vector obtained after the completion of the plurality of matrix multiplications included in the parity generation equation is output to the code inverting unit.   The sum of the number of non-zero elements in the non-diagonal terms of all matrices included in the parity generation formula is equal to or less than the sum of the number of non-zero elements in the non-diagonal terms of all matrices included in the formula (A6). The encoding device according to claim 12. The encoding device is an encoding device that generates a matrix included in the parity generation equation before generating the parity vector Y,
The intermediate result storage unit further stores data representing an intermediate result of processing for generating a matrix included in the parity generation formula,
The matrix product calculation unit further performs multiplication of two or more adjacent matrices included in the formula (A6), and outputs data representing the matrix C as a multiplication result,
The encoding device further includes:
A parity check matrix H that is decomposed into the matrix A and the matrix B, and a parity check matrix decomposition unit that stores data representing the matrix A in the matrix storage unit;
LU which decomposes the matrix B into a product of the lower triangular matrix L and the upper triangular matrix U and stores the data representing the lower triangular matrix L and the data representing the upper triangular matrix U in the intermediate result storage unit A decomposition unit;
Data representing the upper triangular matrix U and the lower triangular matrix L is read from the intermediate result storage unit, and the upper triangular matrix U is converted into a product of m basic upper triangular matrices U _i (i = 1 to m). The lower triangular matrix L is decomposed into a product of m basic lower triangular matrices L _i (i = 1 to m), and the decomposed matrix U _i (i = 1 to m) and the matrix L _i ( a basic matrix decomposition unit that stores data representing i = 1 to m) in the intermediate result storage unit;
Data representing the matrix U _i (i = 1 to m) and the matrix L _i (i = 1 to m) is read from the intermediate result storage unit, and the matrix U _i (i = 1 to m) and the matrix By inverting the sign of the element of the off-diagonal term of L _i (i = 1 to m), the inverse matrix U _i ⁻¹ (i = ^{1 to} m) and the inverse matrix L _i ⁻¹ (i = ^{1 to} m) ) And stores the data representing the inverse matrix U _i ⁻¹ (i = ^{1 to} m) and the inverse matrix L _i ⁻¹ (i = ^{1 to} m) in the intermediate result storage unit. When,
The sum of the number of non-zero elements of the matrix C, which is the multiplication result, and the number of non-zero elements in the off-diagonal terms of the matrix excluding the two or more adjacent matrices among the matrices included in the equation (A6), Among the matrices included in the equation (A6) stored in the intermediate result storage unit when the number of non-diagonal terms of all the matrices included in the equation (A6) is less than or equal to the sum The encoding apparatus according to claim 13, further comprising: a matrix multiplication number reduction unit that stores data representing a matrix excluding the two or more adjacent matrices and data representing the matrix C in the matrix storage unit. . From the information vector X with k elements, the equation (A7-1) is obtained when u ≧ 1 and v ≧ 1, the equation (A7-2) when u ≧ 1 and v = 0, and the equation (A7-2) when u = 0 and v ≧ 1. An operation modulo 2 according to a parity generation equation in which at least one set of two or more adjacent matrices included in equation (A7-3) is replaced by matrix C that is the result of multiplication of the two or more matrices Is a coding device that generates a parity vector Y having m elements by performing
Y≡E ^(v) U ′ ₁ U ′ ₂ ... U ′ _m−1 U ′ _m L ′ _m L ′ _m−1 ... L ′ ₂ L ′ ₁ A ^(u) X (A7 -1)
Y≡U ′ ₁ U ′ ₂ ... U ′ _m-1 U ′ _m L ′ _m L ′ _m−1 ... L ′ ₂ L ′ ₁ A ^(u) X (A7-2)
Y≡E ^(v) U ′ ₁ U ′ ₂ ... U ′ _m−1 U ′ _m L ′ _m L ′ _m−1 ... L ′ ₂ L ′ ₁ AX (A7-3)
However, (X, Y) ^T represents transposition of the vector (X, Y), the parity check matrix H is an m-row n-column matrix satisfying H (X, Y) ^T ≡0, and the matrix A is , The first column to the k-th column of the parity check matrix H, the matrix B includes the (k + 1) -th column to the n-th column of the parity check matrix H, and n = k + m,
When u ≧ 1 and v ≧ 1, the matrix B ′ is obtained by performing u times of row replacement and v times of column replacement for the matrix B, and the matrix A ^(u) Is the same permutation as the u-th row permutation for the matrix B with respect to the matrix A, and the matrix E ^(v) is v times for the matrix B with respect to the unit matrix E. It is the same replacement as the column replacement,
When u ≧ 1 and v = 0, the matrix B ′ is obtained by performing u times of row replacement with respect to the matrix B, and the matrix A ^(u) is obtained with respect to the matrix A. , And the same replacement as the u-time replacement of the matrix B,
When u = 0 and v ≧ 1, the matrix B ′ is obtained by performing v times of column replacement with respect to the matrix B, and the matrix E ^(v) is obtained with respect to the unit matrix E. , The same permutation as the column permutation of the matrix B is performed,
The matrix L ′ _i (i = 1 to m) is a basic lower triangular matrix of the lower triangular matrix L ′ obtained by LU decomposition of the matrix B ′, and the i-th column is the i-th column of the lower triangular matrix L ′. The other column is a matrix whose off-diagonal term is 0 and whose diagonal term is 1,
The matrix U ′ _i (i = 1 to m) is a basic upper triangular matrix of the upper triangular matrix U ′ obtained by LU decomposition of the matrix B ′, and the i-th row is the i-th row of the upper triangular matrix U ′. The other row is a matrix whose off-diagonal terms are 0 and diagonal terms are 1,
The encoding device includes:
A matrix storage unit for storing data representing a plurality of matrices included in the parity generation equation;
An intermediate result storage unit for storing data representing intermediate results of a plurality of matrix multiplications included in the parity generation equation;
In order to calculate the parity vector Y, a matrix product calculation unit that performs a plurality of matrix multiplications included in the parity generation equation,
The matrix product calculation unit performs each matrix multiplication included in the parity generation formula using the information vector X, data representing a matrix in the matrix storage unit, or data representing an intermediate result in the intermediate result storage unit Until the matrix multiplication of the plurality of times included in the parity generation formula ends, data representing the calculation result is stored in the intermediate result storage unit as data representing the intermediate result, and the parity generation formula When the plurality of matrix multiplications included in is completed, a vector obtained after the completion of the plurality of matrix multiplications included in the parity generation equation is output as a parity vector Y.   The sum of the number of non-zero elements of the non-diagonal terms of all the matrices included in the parity generation equation is the sum of all the matrices included in the equation (A7-1), (A7-2), or (A7-3). The encoding device according to claim 15, which is equal to or less than the sum of the number of non-zero elements of the non-diagonal terms. The encoding device is an encoding device that generates a matrix included in the parity generation equation before generating the parity vector Y,
The intermediate result storage unit further stores data representing an intermediate result of processing for generating a matrix included in the parity generation formula,
The matrix product calculation unit further performs multiplication of two or more adjacent matrices included in the formula (A7-1), (A7-2), or (A7-3), and a matrix that is a multiplication result Output data representing C,
The encoding device further includes:
A parity check matrix H is decomposed into the matrix A and the matrix B, and a parity check matrix decomposition unit that stores data representing the matrix A and the matrix B in the intermediate result storage unit;
When u ≧ 1 and v ≧ 1, the data representing the matrix A and the matrix B is read from the intermediate result storage unit, the matrix B is ordered, and u rows are exchanged for the matrix B And the matrix B ′ in which the column replacement is performed v times, and the matrix A ^{(u) in} which the same replacement as the u-time replacement of the matrix B is performed on the matrix A is generated. The matrix E ^(v) is stored in the intermediate result storage unit, and the unit matrix E is exchanged in the same manner as the v-time column replacement for the matrix B to generate the matrix E ^(v) in the intermediate result storage unit. When u ≧ 1 and v = 0, the data representing the matrix A and the matrix B is read from the intermediate result storage unit, the matrix B is ordered, and u times for the matrix B Swap lines Generating the matrix B 'was conducted, the matrix generation and stores the intermediate result storage unit the matrix A was carried out in the same manner as the replacement and swapping of u times the line ^(u) with respect to B with respect to the matrix A If u = 0 and v ≧ 1, the data representing the matrix B is read from the intermediate result storage unit, the matrix B is ordered, and the matrix B is replaced v times. The matrix B ′ is generated, and the matrix E ^(v) obtained by performing the same replacement of the unit matrix E as the column replacement for the matrix B is generated and stored in the intermediate result storage unit. An ordering section to be
The matrix B ′ is LU-decomposed into a product of the lower triangular matrix L ′ and the upper triangular matrix U ′, and data representing the lower triangular matrix L ′ and data representing the upper triangular matrix U ′ are stored in the intermediate result. LU decomposition unit to be stored in the unit,
Data representing the upper triangular matrix U ′ and the lower triangular matrix L ′ is read from the intermediate result storage unit, and the upper triangular matrix U ′ is converted into the m basic upper triangular matrices U ′ _i (i = 1 to m). ), The lower triangular matrix L ′ is decomposed into products of m basic lower triangular matrices L ′ _i (i = 1 to m), and the decomposed matrix U ′ _i (i = 1 to 1) is decomposed. m) and a matrix decomposing unit that stores data representing the matrix 'L _i (i = 1 to m) in the intermediate result storage unit;
The number of non-zero elements of the matrix C, which is the multiplication result, and the two or more adjacent matrices are excluded from the matrices included in the formula (A7-1), (A7-2), or (A7-3). The sum of the number of non-zero elements in the off-diagonal terms of the matrix is the non-zero element in the off-diagonal terms of all the matrices included in the formula (A7-1), (A7-2), or (A7-3). Two adjacent ones of the matrices included in the formula (A7-1), (A7-2), or (A7-3) stored in the intermediate result storage unit when the sum is less than or equal to the sum of numbers 17. The encoding apparatus according to claim 16, further comprising: a matrix multiplication number reduction unit that stores data representing a matrix excluding the matrix and data representing the matrix C in the matrix storage unit. From the information vector X with k elements, the equation (A8-1) is obtained when u ≧ 1 and v ≧ 1, the equation (A8-2) when u ≧ 1 and v = 0, and the equation (A8-2) when u = 0 and v ≧ 1. By performing a real number operation according to a parity generation equation in which at least one set of two or more adjacent matrices included in the equation (A8-3) is replaced with a matrix C that is a multiplication result of the two or more matrices An encoding device for generating a parity vector Y having m elements,
^{Y = -E (v) U '} 1 -1 U' 2 -1 ··· U 'm-1 -1 U' m -1 L 'm -1 L' m-1 -1 ··· L '2 ^-1 L ' ₁ ^-1 A ^(u) X (A8-1)
Y = -U ' ₁ ^-1 U' ₂ ^-1 ... U ' _m-1 ^-1 U' _m ^-1 L ' _m ^-1 L' _m-1 ^-1 ... L ' ₂ ^-1 L' ₁ ^-1 A ^(V) X (A8-2)
^{Y = -E (v) U '} 1 -1 U' 2 -1 ··· U 'm-1 -1 U' m -1 L 'm -1 L' m-1 -1 ··· L '2 ^-1 L ' ₁ ^-1 AX (A8-3)
However, (X, Y) ^T represents transposition of the vector (X, Y), the parity check matrix H is an m-by-n matrix satisfying H (X, Y) ^T = 0, and the matrix A is , The first column to the k-th column of the parity check matrix H, the matrix B includes the (k + 1) -th column to the n-th column of the parity check matrix H, and n = k + m,
When u ≧ 1 and v ≧ 1, the matrix B ′ is obtained by performing u times of row replacement and v times of column replacement for the matrix B, and the matrix A ^(u) Is the same permutation as the u-th row permutation for the matrix B with respect to the matrix A, and the matrix E ^(v) is v times for the matrix B with respect to the unit matrix E. It is the same replacement as the column replacement,
When u ≧ 1 and v = 0, the matrix B ′ is obtained by performing u times of row replacement with respect to the matrix B, and the matrix A ^(u) is obtained with respect to the matrix A. , And the same replacement as the u-time replacement of the matrix B,
When u = 0 and v ≧ 1, the matrix B ′ is obtained by performing v times of column replacement with respect to the matrix B, and the matrix E ^(v) is obtained with respect to the unit matrix E. , The same permutation as the column permutation of the matrix B is performed,
Matrix ^{_{L 'i -1 (i = 1~m}} ) is a matrix L' is the inverse matrix of _i, the matrix L _'i, said matrix B' of the basic lower triangular matrix of LU decomposition lower triangular matrix has L ' The i-th column matches the i-th column of the lower triangular matrix L ′, and the other columns are matrices whose off-diagonal terms are 0 and whose diagonal terms are 1.
Matrix ^{_{U 'i -1 (i = 1~m}} ) is a matrix U' is the inverse matrix of _i, the matrix U _'i, said matrix B' of the basic upper triangular matrix of the upper triangular matrix U 'was LU decomposition The i-th row coincides with the i-th row of the upper triangular matrix U ′, and the other rows are matrices in which the off-diagonal term is 0 and the diagonal term is 1.
The encoding device includes:
A matrix storage unit for storing data representing a plurality of matrices included in the parity generation equation;
An intermediate result storage unit for storing data representing intermediate results of a plurality of matrix multiplications included in the parity generation equation;
In order to calculate the parity vector Y, a matrix product calculator that performs a plurality of matrix multiplications included in the parity generation equation;
A code inversion unit that inverts the sign of the vector output from the matrix product calculation unit and outputs it as a parity vector Y,
The matrix product calculation unit performs each matrix multiplication included in the parity generation formula using the information vector X, data representing a matrix in the matrix storage unit, or data representing an intermediate result in the intermediate result storage unit Until the matrix multiplication of the plurality of times included in the parity generation formula ends, data representing the calculation result is stored in the intermediate result storage unit as data representing the intermediate result, and the parity generation formula When the plurality of matrix multiplications included in is completed, a vector obtained after the completion of the plurality of matrix multiplications included in the parity generation equation is output to the code inverting unit.   The sum of the number of non-zero elements of the non-diagonal terms of all the matrices included in the parity generation equation is the sum of all the matrices included in the equation (A8-1), (A8-2), or (A8-3). The encoding device according to claim 18, which is equal to or less than the sum of the number of non-zero elements of the non-diagonal terms. The encoding device is an encoding device that generates a matrix included in the parity generation equation before generating the parity vector Y,
The intermediate result storage unit further stores data representing an intermediate result of processing for generating a matrix included in the parity generation formula,
The matrix product calculation unit further performs multiplication of two or more adjacent matrices included in the formula (A8-1), (A8-2), or (A8-3), and a matrix that is a multiplication result Output data representing C,
The encoding device further includes:
A parity check matrix H is decomposed into the matrix A and the matrix B, and a parity check matrix decomposition unit that stores data representing the matrix A and the matrix B in the intermediate result storage unit;
When u ≧ 1 and v ≧ 1, the data representing the matrix A and the matrix B is read from the intermediate result storage unit, the matrix B is ordered, and u rows are exchanged for the matrix B And the matrix B ′ in which the column replacement is performed v times, and the matrix A ^{(u) in} which the same replacement as the u-time replacement of the matrix B is performed on the matrix A is generated. The matrix E ^(v) is stored in the intermediate result storage unit, and the unit matrix E is exchanged in the same manner as the v-time column replacement for the matrix B to generate the matrix E ^(v) in the intermediate result storage unit. When u ≧ 1 and v = 0, the data representing the matrix A and the matrix B is read from the intermediate result storage unit, the matrix B is ordered, and u times for the matrix B Swap lines Generating the matrix B 'was conducted, the matrix generation and stores the intermediate result storage unit the matrix A was carried out in the same manner as the replacement and swapping of u times the line ^(u) with respect to B with respect to the matrix A If u = 0 and v ≧ 1, the data representing the matrix B is read from the intermediate result storage unit, the matrix B is ordered, and the matrix B is replaced v times. The matrix B ′ is generated, and the matrix E ^(v) obtained by performing the same replacement of the unit matrix E as the column replacement for the matrix B is generated and stored in the intermediate result storage unit. An ordering section to be
The matrix B ′ is LU-decomposed into a product of the lower triangular matrix L ′ and the upper triangular matrix U ′, and data representing the lower triangular matrix L ′ and data representing the upper triangular matrix U ′ are stored in the intermediate result. LU decomposition unit to be stored in the unit,
Data representing the upper triangular matrix U ′ and the lower triangular matrix L ′ is read from the intermediate result storage unit, and the upper triangular matrix U ′ is converted into the m basic upper triangular matrices U ′ _i (i = 1 to m). ), The lower triangular matrix L ′ is decomposed into products of m basic lower triangular matrices L ′ _i (i = 1 to m), and the decomposed matrix U ′ _i (i = 1). ˜m) and matrix ′ L _i (i = 1 to m) are stored in the intermediate result storage unit, a basic matrix decomposition unit;
The intermediate results from the storage unit reads the data representing the matrix U _{'i (i} = 1~m) and the matrix _{L' i (i = 1~m)} , the matrix U _{'i (i} = 1~m) And the inverse matrix U ′ _i ⁻¹ (i = ^{1 to} m) and the inverse matrix L ′ _i ⁻¹ by inverting the signs of the elements of the off-diagonal terms of the matrix L ′ _i (i = ^{1 to} m). (I = 1 to m) is calculated, and data representing the inverse matrix U ′ _i ⁻¹ (i = ^{1 to} m) and the inverse matrix L ′ _i ⁻¹ (i = ^{1 to} m) is stored as the intermediate result. An inverse matrix calculation unit to be stored in the unit,
The number of non-zero elements of the matrix C which is the multiplication result and the two or more adjacent matrices are excluded from the matrices included in the formula (A8-1), (A8-2), or (A8-3). The sum of the number of non-zero elements of the non-diagonal terms of the matrix is the non-zero element of the non-diagonal terms of all the matrices included in the formula (A8-1), (A8-2), or (A8-3). The adjacent two of the matrices included in the formula (A8-1), (A8-2), or (A8-3) stored in the intermediate result storage unit when the sum is less than or equal to the sum of the numbers The encoding apparatus according to claim 19, further comprising: a matrix multiplication number reduction unit that stores data representing a matrix excluding the matrix and data representing the matrix C in the matrix storage unit.   The matrix product calculation unit sequentially executes a matrix multiplication on the right side of the parity generation formula among a plurality of matrix multiplications included in the parity generation formula. , 15 and 18.   The matrix product calculation unit performs two or more matrix multiplications in parallel among a plurality of matrix multiplications included in the parity generation formula, in parallel. The encoding device according to any one of 18. An encoding program for generating a parity vector Y having m elements by performing an operation modulo 2 from an information vector X having k elements in accordance with a parity generation expression represented by Expression (A9),
Y≡U ₁ U ₂ ... U _m-1 U _m L _m L _m-1 ... L ₂ L ₁ AX (A9)
However, (X, Y) ^T represents transposition of the vector (X, Y), the parity check matrix H is an m-row n-column matrix satisfying H (X, Y) ^T ≡0, and the matrix A is , The first column to the k-th column of the parity check matrix H, the matrix B includes the (k + 1) -th column to the n-th column of the parity check matrix H, and n = k + m,
The matrix L _i (i = 1 to m) is a basic lower triangular matrix of the lower triangular matrix L obtained by LU decomposition of the matrix B, and the i-th column matches the i-th column of the lower triangular matrix L. , The other columns are matrices with off-diagonal terms 0 and diagonal terms 1.
The matrix U _i (i = 1 to m) is a basic upper triangular matrix of the upper triangular matrix U obtained by LU decomposition of the matrix B, and the i-th row matches the i-th row of the upper triangular matrix U. , The other rows are matrices with off-diagonal terms of 0 and diagonal terms of 1,
The encoding program includes a computer,
A matrix storage unit for storing data representing a plurality of matrices included in the parity generation equation;
An intermediate result storage unit for storing data representing intermediate results of a plurality of matrix multiplications included in the parity generation equation;
In order to calculate the parity vector Y, function as a matrix product calculation unit that performs a plurality of matrix multiplications included in the parity generation formula,
The matrix product calculation unit performs each matrix multiplication included in the parity generation formula using the information vector X, data representing a matrix in the matrix storage unit, or data representing an intermediate result in the intermediate result storage unit Until the matrix multiplication of the plurality of times included in the parity generation formula ends, data representing the calculation result is stored in the intermediate result storage unit as data representing the intermediate result, and the parity generation formula When the plurality of matrix multiplications included in is completed, a vector obtained after the completion of the plurality of matrix multiplications included in the parity generation equation is output as a parity vector Y. The encoding program is an encoding program for generating a matrix included in the parity generation formula before generating the parity vector Y,
The intermediate result storage unit further stores data representing an intermediate result of processing for generating a matrix included in the parity generation formula,
The encoding program further includes the computer,
A parity check matrix H that is decomposed into the matrix A and the matrix B, and a parity check matrix decomposition unit that stores data representing the matrix A in the matrix storage unit;
LU which decomposes the matrix B into a product of the lower triangular matrix L and the upper triangular matrix U and stores the data representing the lower triangular matrix L and the data representing the upper triangular matrix U in the intermediate result storage unit A decomposition unit;
Data representing the upper triangular matrix U and the lower triangular matrix L is read from the intermediate result storage unit, and the upper triangular matrix U is converted into a product of m basic upper triangular matrices U _i (i = 1 to m). The lower triangular matrix L is decomposed into a product of m basic lower triangular matrices L _i (i = 1 to m), and the decomposed matrix U _i (i = 1 to m) and the matrix L _i are decomposed. 24. The encoding program according to claim 23, wherein the encoding program causes a function of a basic matrix decomposition unit to store data representing (i = 1 to m) in the matrix storage unit.

Images