JP5005743B2 - Encoding device, decoding device, encoding method, decoding method, encoding program, and decoding program - Google Patents

Description

  The present invention relates to encoding for generating a constant type code having a constant Hamming weight and decoding thereof, and particularly to encoding and decoding for efficiently generating a constant type code regardless of the length of a codeword. The present invention relates to an encoding device, a decoding device, an encoding method, a decoding method, an encoding program, and a decoding program that can be performed.

  Conventionally, when a channel code or a distorted information source code has a property of asymptotically approaching the Shannon limit with respect to a codeword length (block length), typical sequences are generally used as these codes (for example, (See Patent Document 1 and Patent Document 2). Here, the typical sequence refers to a sequence in which the Hamming weights of codewords are distributed around a predetermined constant value, that is, the Hamming weight has a width. The Hamming weight indicates the number of symbols other than 0 in the code word.

  By the way, a code word having a constant Hamming weight (hereinafter referred to as “constant composition code”) is used to prove reachability to the Shannon limit (for example, non-patent literature). 1), which is often used in theoretical analysis.

  The fixed type code is also used for radio system technology (for example, see Non-Patent Document 2) and search technology (for example, see Non-Patent Document 3). Non-Patent Document 4 discloses a technique for converting a constraint condition on a finite field into a linear constraint condition on a real field.

JP 2008-177892 A JP 2009-060288 A

I. Csiszar and J. Korner, Information Theory: Coding Theorems for Discrete Memoryless Systems, Academic Press, 1981 D. H. Smith, L. A. Hughes and S. Perkins, "A New Table of Constant Weight Codes of Length Greater than 28," The Electronic Journal of Combinatorics 13, 2006 Miyake, “Information Theoretic Approach to Index Structure of Full-Text Search System,” IEICE Technical Report, PRMU2008-108 (IT-2008-65), March 2009, P137-141 J. Feldman, M. J. Wainwright, and D. R. Karger, “Using linear programming to decode binary linear codes,” IEEE Trans. Inform. Theory, vol. 51, pp. 954-972, 2005

  However, it is not yet clear how to actually configure the constant type code. When the code word length (block length) n is smaller than 100, the constant type code is generated according to a table prepared in advance. (For example, refer nonpatent literature 2).

  That is, the conventional type constant code is used in the field of theoretical analysis, but its application range is limited to a case where the code word length (block length) n is small, and the application field is limited. was there.

  In addition, when the codeword length (block length) n increases, the conventional constant type code generation method increases the amount of calculation exponentially with respect to n. Therefore, the constant type code corresponding to a long codeword length is efficiently used. It was difficult to produce well.

  Therefore, an encoding device, a decoding device, an encoding method, a decoding method, a code capable of efficiently generating a constant type code regardless of the length of the codeword and the decoding thereof can be performed. How to implement the encryption program or the decryption program is a big issue.

  The present invention has been made to solve the above-described problems caused by the prior art, and can perform encoding and decoding to efficiently generate a constant type code regardless of the length of the codeword. An object is to provide an encoding device, a decoding device, an encoding method, a decoding method, an encoding program, and a decoding program.

In order to solve the above-described problems and achieve the object, the present invention is an encoding device that generates a channel code having a constant Hamming weight from a binary message, and includes a first sparse matrix and a second sparse matrix. A storage means for storing a matrix, a one-row row matrix having a predetermined Hamming weight, and a predetermined type; and a one-row row matrix having a product of a codeword and the first sparse matrix having the predetermined Hamming weight ; A first constraint that is equal, a second constraint that a product of the codeword and the second sparse matrix is equal to the binary message, and the code of the codeword The first constraint equation, the first constraint equation, wherein the Hamming weight, which is the product of the word length and the type, is equal to the number of digits in which the codeword element is 1, 2 constraint equations and the third constraint equation are relaxed to a linear constraint on a real field. After having converted to, it is characterized in that an optimization execution means for solving the code word that satisfies all the linear constraints in a linear programming.

Further, the present invention is a decoding device that generates a binary message before encoding from a channel output, and includes a first sparse matrix, a second sparse matrix, and a predetermined Hamming weight used for encoding. A row having a row matrix and a predetermined type, and a product of a coded word representing the encoded binary message and the first sparse matrix has a row with the predetermined Hamming weight. A first constraint equation that is equal to a row matrix , the Hamming weight that is the product of the codeword length of the codeword and the type is equal to the number of digits in which the codeword element is 1 A second constraint equation, a third constraint equation in which the number of digits in which both the codeword element and the channel output element are both equal to a predetermined natural number, and Using a linear objective function for the codeword, the first constraint equation, the first The code word that minimizes the objective function among the code words that satisfy all of the linear constraint conditions after the constraint formula and the third constraint formula are converted into linear constraint conditions on the real field by a relaxation method And an optimization execution unit that uses a product of the code word and the second sparse matrix as the binary message.

The present invention, in the above invention, the objective function used by the optimization execution means, the codeword x ^n, wherein a communication channel output y ^n, the code word length n, the type P, hb ([delta]) and _{H b (δ) = - δlogδ-} (1-δ) log (1-δ), sequence ^r the number of occurrences of a in ^{n n (a | r n)} , the predetermined natural number m 0 ≦ m ≦ min {nP, N (1 | y ⁿ )}, (N (0 | y ⁿ ) / n) H _b ((nP−m) / N (0 | y ⁿ )) + ( N (1 | y ⁿ ) / n) H _b (m / N (1 | y ⁿ )).

  Further, the present invention is the above invention, wherein the predetermined natural number is not less than 0 and not more than a smaller value of the number of digits in which the Hamming weight or the channel output element is 1. Whether the optimization execution unit has an executable solution for the codeword that satisfies all of the first constraint equation, the second constraint equation, and the third constraint equation for each of the predetermined natural numbers. It is determined whether or not, and when it is determined that the feasible solution exists, the value of the objective function is calculated.

The present invention also relates to an encoding device that generates a distorted information source code having a constant Hamming weight based on a decoder output obtained from an information source output, the first sparse matrix and the second sparse matrix. A storage means for storing a matrix, a row matrix having a predetermined Hamming weight, and a predetermined type, and a product of the decoder output and the first sparse matrix is a row having the predetermined Hamming weight. first constraint expression that is a lateral matrix and equal value, the number the Hamming weight of the digit elements of the decoder output is 1 is the product of the codeword length and the type of the decoder output A third constraint expression that is equal to a predetermined natural number, a second constraint expression that is equal to the second natural number, and that the number of digits in which both the information source output element and the decoder output element are both 1 And a linear objective function with respect to the decoder output, Of the decoder outputs satisfying all the linear constraint conditions after converting the constraint formula of 1, the second constraint formula and the third constraint formula into linear constraint conditions on a real field by a relaxation method, Optimization execution means for solving the decoder output that minimizes the objective function by linear programming and using a product of the decoder output and the second sparse matrix as the distorted information source code. It is characterized by that.

Further, in the present invention, the objective function used by the optimization execution means in the above invention is that the information source output is x ⁿ , the decoder output is y ⁿ , the codeword length is n, and the type is P, p is 0 <p <1, q is 0 <q <1, D (p‖q) is D (p‖q) = plog (p / q) + (1-p) log ((1-p ) / (1-q)), where N (a | r ⁿ ) is the number of occurrences of a in the sequence r ⁿ and W _{Y | X} is a predetermined conditional probability, (N (0 | y ⁿ ) / n) D ((nP−m) / N (0 | x ⁿ ) ‖W _{Y | X} (1 | 0)) + (N (1 | y ⁿ ) / n) D (m / N (1 | y ⁿ ) W _{Y | X} (1 | 1)).

  Further, the present invention is the above invention, wherein the predetermined natural number is not less than 0 and not more than the smaller value of the number of digits in which the Hamming weight or the element of the information source output is 1. The optimization execution means has an executable solution for the decoder output that satisfies all of the first constraint equation, the second constraint equation, and the third constraint equation for each of the predetermined natural numbers. It is determined whether or not to execute, and when it is determined that the executable solution exists, the value of the objective function is calculated.

The present invention is also a decoding device that generates a decoder output from a distorted information source code, and has a first sparse matrix, a second sparse matrix, and a predetermined Hamming weight used for encoding. A storage means for storing a row matrix of a row and a predetermined type, and a product of the decoder output and the first sparse matrix is equal to a row matrix of the row having the predetermined Hamming weight A second constraint equation that the product of the decoder output and the second sparse matrix is equal to the distorted information source code, and the decoder output Using the third constraint equation, wherein the Hamming weight, which is the product of the codeword length and the type, is equal to the number of digits whose elements of the decoder output are 1, the first constraint equation , Converting the second constraint equation and the third constraint equation into linear constraint conditions on a real field by a relaxation method In Ue, characterized in that a optimized execution means for solving the decoder output that satisfies all the linear constraints in a linear programming.

In addition, the present invention is an encoding method for generating a channel code having a constant Hamming weight from a binary message, and includes a first sparse matrix, a second sparse matrix, and a row having a predetermined Hamming weight. A storage step of storing a horizontal matrix and a predetermined type in the storage unit, and a first product that is a product of a codeword and the first sparse matrix is equal to a one-row horizontal matrix having the predetermined Hamming weight , A second constraint expression that the product of the codeword and the second sparse matrix is equivalent to the binary message, and a product of the codeword length of the codeword and the type The Hamming weight is equal to the number of digits in which the codeword element is 1, and the third constraint equation is used, and the first constraint equation, the second constraint equation, and the third constraint equation are used. Is converted into a linear constraint on the real field by the relaxation method, The code word that satisfies all about conditions characterized by containing an optimized execution step of solving a linear programming.

The present invention is also a decoding method for generating a binary message before encoding from a communication channel output, wherein a first sparse matrix, a second sparse matrix, and a predetermined Hamming weight used for encoding are obtained. The product of the storage step of storing the row matrix of one row and the predetermined type in the storage unit, the code word representing the encoded binary message and the first sparse matrix has the predetermined Hamming weight A first constraint equation that is equal to a row matrix of one row, the Hamming weight that is the product of the codeword length of the codeword and the type, and the number of digits in which the codeword element is 1 A second constraint expression that is equal, and a third constraint expression that the number of digits in which both the codeword element and the channel output element are both equal to a predetermined natural number And the first constraint using a linear objective function with respect to the codeword The second constraint equation and the third constraint equation are converted into linear constraint conditions on a real field by a relaxation method, and the objective function is minimized among the codewords that satisfy all the linear constraint conditions. An optimization execution step in which the codeword is solved by linear programming and a product of the codeword and the second sparse matrix is used as the binary message.

The present invention also relates to an encoding method for generating a distorted information source code having a constant Hamming weight based on a decoder output obtained from an information source output, the first sparse matrix and the second sparse matrix. A storage step of storing a matrix, a one-row horizontal matrix having a predetermined Hamming weight and a predetermined type in a storage unit, and a product of the decoder output and the first sparse matrix has the predetermined Hamming weight first constraint expression that is a lateral matrix and equal values of one row, the Hamming weight of the decoder output element is the product of the codeword length and the type of the decoder output is 1 A second constraint expression that is equal to the number of digits, the number of digits in which both the information source output element and the decoder output element are both equal to a predetermined natural number And a linear objective function with respect to the decoder output The decoder that satisfies all of the linear constraint conditions after converting the first constraint formula, the second constraint formula, and the third constraint formula into linear constraint conditions on a real field by a relaxation method. Among the outputs, the decoder output that minimizes the objective function is solved by linear programming, and the product of the decoder output and the second sparse matrix is used as the distortion information source code And an execution step.

The present invention is also a decoding method for generating a decoder output from a distorted information source code, which has a first sparse matrix, a second sparse matrix, and a predetermined Hamming weight used for encoding. A storage step of storing a row of a matrix and a predetermined type in a storage unit, and a product of the decoder output and the first sparse matrix is equal to a row of a matrix having the predetermined Hamming weight A second constraint equation that the product of the decoder output and the second sparse matrix is equal to the distorted information source code, and the decoding A third constraint equation that assumes that the Hamming weight, which is the product of the codeword length of the decoder output and the type, is equal to the number of digits whose elements of the decoder output are 1; , The second constraint equation and the third constraint equation are linearly constrained on a real field by a relaxation method After having converted into matter, characterized in that they contain and optimization execution step of solving the decoder output that satisfies all the linear constraints in a linear programming.

According to the present invention, a first sparse matrix, a second sparse matrix, a row matrix having a predetermined Hamming weight and a predetermined type are stored, and a product of the codeword and the first sparse matrix is predetermined. A first constraint expression that is equal to a row matrix of one row having a Hamming weight of 2 and a second constraint expression that a product of a codeword and a second sparse matrix is equal to a binary message , And a third constraint equation that assumes that the Hamming weight, which is the product of the codeword length and the type of the codeword, is equal to the number of digits in which the codeword element is 1, and the first constraint equation Since the second constraint equation and the third constraint equation are converted into linear constraints on the real number field by the relaxation method, codewords that satisfy all of these linear constraints are solved by linear programming. By considering each constraint as a linear constraint on the real field, it is effective regardless of the codeword length. To an effect that it is possible to realize the coding to generate the type constant sign. Further, the transmission capacity closer to the Shannon limit can be realized than the code using the typical sequence.

In addition, according to the present invention, the first sparse matrix, the second sparse matrix used for encoding, the one-row horizontal matrix having a predetermined Hamming weight, and a predetermined type are stored and encoded 2 A first constraint expression that the product of a codeword representing a value message and a first sparse matrix is equal to a row matrix having a predetermined Hamming weight, and the codeword length and type of the codeword The second constraint expression that the Hamming weight as the product is equal to the number of digits whose codeword element is 1, the number of digits whose codeword element and channel output element are both 1 A third constraint equation that is equal to a predetermined natural number, and a linear objective function with respect to the codeword, and the first constraint equation, the second constraint equation, and the third constraint equation are reduced by a relaxation method A code that satisfies all of these linear constraints after conversion to linear constraints on the real field Among them, the codeword that minimizes the objective function is solved by linear programming, and the product of the solved codeword and the second sparse matrix is set as a binary message. By using a linear objective function, it is possible to efficiently decode a constant type code.

In addition, according to the present invention, the objective function has the code word x ⁿ , the channel output y ⁿ , the code word length n, the type P, and Hb (δ) as H _b (δ) = − δlogδ− ( 1-δ) log (1-δ), the number of occurrences of a in the sequence r ⁿ is N (a | r ⁿ ), and the predetermined natural number m is 0 ≦ m ≦ min {nP, N (1 | y ⁿ )}. (N (0 | y ⁿ ) / n) H _b ((nP−m) / N (0 | y ⁿ )) + (N (1 | y ⁿ ) / n) H _b (m / N (1 | y ⁿ )), the constant type code can be efficiently decoded.

  Further, according to the present invention, the predetermined natural number is 0 or more and not more than the smaller value of the number of digits in which the Hamming weight or channel output element is 1, and for each predetermined natural number , It is determined whether there is an executable solution for a codeword that satisfies all of the first constraint equation, the second constraint equation, and the third constraint equation, and if it is determined that there is an executable solution, Since the function value is calculated, the processing load related to decoding can be reduced.

In addition, according to the present invention, a first sparse matrix, a second sparse matrix, a row matrix having a predetermined Hamming weight, and a predetermined type are stored, and the decoder output and the first sparse matrix horizontal matrix and the first constraint to be equal values, the Hamming weight of the decoder output is the product of the codeword length and type of decoder output product of one line having a predetermined Hamming weight The second constraint expression that the number of digits whose elements are 1 is equal to the number of digits, the number of digits whose both the source output element and the decoder output element are both equal to a predetermined natural number And a linear objective function with respect to the decoder output, and the first constraint equation, the second constraint equation, and the third constraint equation are linear on the real field by the relaxation method. Of the decoder outputs that satisfy all of these linear constraints after conversion to constraints, the objective function Since the decoder output to be minimized is solved by linear programming, and the product of the obtained decoder output and the second sparse matrix is used as a distorted information source code, each constraint expression is expressed in a real field. By using a linear objective function, it is possible to realize an encoding that can efficiently generate a constant type code regardless of the length of the codeword.

In addition, according to the present invention, the objective function includes the source output x ⁿ , the decoder output y ⁿ , the codeword length n, the type P, p 0 <p <1, and q 0 <q. <1, D a (p‖q) D (p‖q) = plog (p / q) + (1-p) log ((1-p) / (1-q)), the a in the sequence ^{r n} When the number of appearances is N (a | r ⁿ ) and the predetermined conditional probability is W _{Y | X} , (N (0 | y ⁿ ) / n) D ((nP−m) / N (0 | x ⁿ ) ‖W _{Y | X} (1 | 0)) + (N (1 | y ⁿ ) / n) D (m / N (1 | y ⁿ ) ‖W _{Y | X} (1 | 1)) As a result, there is an effect that the encoding for generating the constant type code can be performed efficiently.

  Further, according to the present invention, the predetermined natural number is not less than 0 and not more than the smaller value of the number of digits whose Hamming weight or information source output element is 1, and for each predetermined natural number If there is a feasible solution for the decoder output that satisfies all of the first constraint equation, the second constraint equation, and the third constraint equation, and if it is determined that there is an executable solution Since the value of the objective function is calculated, it is possible to reduce the processing load related to encoding.

Further, according to the present invention, the first sparse matrix, the second sparse matrix used for encoding, the one-row horizontal matrix having a predetermined Hamming weight, and the predetermined type are stored, and the decoder output The first constraint expression that the product of the first sparse matrix is equal to the row matrix of one row having a predetermined Hamming weight, the product of the decoder output and the second sparse matrix is the distortion information The second constraint equation that is equal to the source code, and the Hamming weight that is the product of the codeword length and the type of the decoder output is the number of digits in which the element of the decoder output is 1, etc. The first constraint equation, the second constraint equation, and the third constraint equation are converted into linear constraints on the real number field by the relaxation method using the third constraint equation that is a value, and then these linear Since the decoder output that satisfies all the constraints is solved by linear programming, each constraint expression is linear on the real field. By regarded as about conditions, an effect that it is possible to efficiently decode the type constant sign.

FIG. 1 is a diagram illustrating a channel coding problem and a distortion information source coding problem. FIG. 2 is a diagram showing an outline of the encoding method and the decoding method according to the present invention. FIG. 3 is a block diagram illustrating a configuration of an encoding device related to a channel code. FIG. 4 is a flowchart showing a processing procedure executed by the encoding device relating to the channel code. FIG. 5 is a block diagram showing a configuration of a decoding apparatus related to channel codes. FIG. 6 is a flowchart showing a processing procedure executed by the decoding apparatus related to the channel code. FIG. 7 is a block diagram showing a configuration of an encoding apparatus related to a distorted information source code. FIG. 8 is a flowchart showing a processing procedure executed by the encoding apparatus related to the distorted information source code. FIG. 9 is a block diagram illustrating a configuration of a decoding apparatus related to a distorted information source code. FIG. 10 is a flowchart showing a processing procedure executed by the decoding apparatus related to the distorted information source code.

  Exemplary embodiments of an encoding device, a decoding device, an encoding method, a decoding method, an encoding program, and a decoding program according to the present invention will be described below in detail with reference to the accompanying drawings. In the following, the channel code problem and the distorted information source code problem to which the present invention is applied will be described with reference to FIG. 1 and the outline of the present invention will be described with reference to FIG. An example will be described with reference to FIGS.

  First, the channel code problem and the distorted information source code problem to which the present invention is applied will be described with reference to FIG. FIG. 1 is a diagram illustrating a channel code problem and a distorted information source code problem. Note that (A) in the figure shows the channel code problem, and (B) in the figure shows the distortion information source code problem. The channel code is sometimes called an error correction code, and the distortion information source code is sometimes called vector quantization.

  First, the problem of the channel code problem shown in FIG. Similar problems exist for the strained information source code.

As shown in FIG. 1A, the encoder (Φ _n ) as the sender encodes the message u ^k ε {0, 1} ^k and sends the code word x ⁿ to the communication path. Communication path, the codeword x ⁿ which is accepted as input and outputs as the channel output y ^n. Here, the communication path is expressed as a conditional probability W _{Y | X} (in this case, the description of the subscript n of Y and X is omitted). The decoder (Ψ _n ), which is the receiver, receives the channel output y ⁿ and decodes the restoration message (u ^k with (^: hat symbol)) corresponding to the message u ^k .

  Here, a case where a fixed type code is introduced into the system shown in FIG. The constant type code indicates a code word having a Hamming weight nP, where P (0 <P <1) is a given type and n is the code word length of each code word.

On the other hand, the channel capacity C of the binary symmetric channel BSC (δ) having a crossover probability of δ is
C = 1−Hb (δ)
However, Hb (δ) = − δlogδ− (1-δ) log (1-δ)
It is known to be given by

Therefore, the constant type code is generated by selecting 2 ^nC code words from _n C _nP code word candidates. However, it is clear that the amount of calculation required for such codeword selection operation increases exponentially with respect to the codeword length n. For this reason, conventionally, a constant type code having an arbitrary codeword length n cannot be efficiently calculated.

  Therefore, the present invention provides an encoding method for efficiently generating a constant type code of an arbitrary codeword length n and a decoding method for efficiently decoding a code encoded by such an encoding method. It was.

  Here, the outline | summary of the encoding method and decoding method based on this invention is demonstrated using FIG. In addition, (A-1) in the figure shows an outline of the encoding method related to the channel code, and (A-2) in the same figure shows an outline of the decoding method related to the channel code. .

As shown in (A-1) of FIG. 2, in the encoding method according to the present invention,
Constraint formula: Σx _i = nP
However, Σx _i constitutes an encoder Φn that includes the sum of x _i from i = 1 to n as one of the constraint conditions. Here, such a constraint expression means that the number of digits in which each element x _i of the code word x ⁿ is 1 is equal to the Hamming weight nP.

  As described above, the constraint equation shown in (A-1) of FIG. 2 indicates that the Hamming weight of the code word is constant, that is, the code word is a constant type code. In the past, there was no encoder that included a symbol as a constraint.

Even if the constraint expression that the Hamming weight of the codeword is constant is one of the constraint conditions, in the first place, the computation on the field F ₂ in which the matrix operation in each constraint condition is a binary field Therefore, it is considered that it was recognized that it was impossible to obtain the codeword ^xn satisfying all the constraint conditions by linear calculation.

However, in the encoding method according to the present invention, the constraint condition is converted into a linear constraint condition using a relaxation method (see Non-Patent Document 4 described above), which is a technique for converting the constraint condition into a linear constraint condition on a real field. The codeword ^{xn is determined} by linear programming.

  Therefore, according to the encoding method according to the present invention, it is possible to suppress the calculation amount of a constant type code, which was conventionally considered to increase exponentially with respect to the codeword length n, to a polynomial order calculation amount. That is, even when the codeword length n is increased, the processing load required for generating the constant type code does not increase exponentially with respect to the codeword length n. For this reason, it is possible to efficiently generate a constant type code regardless of the length of the codeword length n.

Further, as shown in (A-2) of FIG. 2, in the decoding method according to the present invention,
Constraint formula: Σx _i = nP
However, Σx _i constitutes a decoder Ψ _n including the sum of x _i from i = 1 to n as one of the constraint conditions.

Then, all constraint conditions are converted into linear constraint conditions using the above-described relaxation method. Under the linear constraint conditions, a codeword ^xn that minimizes a newly derived linear objective function is obtained by linear programming. I decided to solve it. Also, an estimated message (corresponding to the message before encoding) is calculated by multiplying the solved codeword ^xn by a predetermined sparse matrix B. Details of the objective function will be described later.

  Therefore, according to the decoding method according to the present invention, it is possible to efficiently decode a constant type code even when the codeword length n is large. A detailed description of the channel code of the present invention will be given in Example 1 described later.

Returning to the description of FIG. 1, the problem of the distorted information source code problem shown in FIG. As shown in FIG. 1B, the information source is given by the probability P _X (here, the description of the subscript n of X is omitted). Then, the encoder [Phi _n has received the information source output x ⁿ from the information source, the information source output x ⁿ to perform compression operations by encoding the u ^k. Moreover, the decoder [psi _n is a recipient, a code word u ^k received, decrypts the recovered message y ^n. It is also possible to restore the message y ^n, it referred to as a decoder output.

Then, if the distortion measure is “d: {0, 1} × {0, 1} → [0, ∞]”, the information source output x ⁿ and the decoder output y ⁿ are given by the distortion criterion D given in advance. (D> 0)
(1 / n) Σd (x _i , y _i ) <D
However, Σd (x _i , y _i ) is required to satisfy the sum of d (x _i , y _i ) from i = 1 to n.

  Here, even when a constant type code is introduced into the system shown in FIG. 1B, the same problem as in the system shown in FIG. That is, as the codeword length n is increased, the processing load required to generate the constant type code is exponentially increased with respect to the codeword length n.

  Therefore, in the present invention, with respect to the distorted information source code, an encoding method for efficiently generating a constant type code of an arbitrary codeword length n and a code encoded by the encoding method are efficiently decoded. It was decided to provide a decoding method. A detailed description of the distorted information source code of the present invention will be given in Example 2 described later.

  In the first embodiment, a case where the present invention is applied to a channel code will be described. First, the configuration of the encoding device will be described with reference to FIG. FIG. 3 is a block diagram illustrating a configuration of the encoding device 10 relating to the channel code.

  As illustrated in FIG. 3, the encoding device 10 includes an input unit 11, a storage unit 12, a control unit 13, and an output unit 14. The storage unit 12 stores a sparse matrix 12a, a vector 12b, and a type 12c, and the control unit 13 further includes an optimization execution unit 13a.

The input unit 11 is a device that receives a message u ^k (u ^k ε {0, 1} ^k ) to be encoded and performs a process of passing the received message u ^k to the control unit 13. The output unit 14 is a device that performs processing for receiving the codeword ^xn from the control unit 13 and outputting the received codeword ^xn to a communication path (not shown).

  The storage unit 12 is a storage unit configured by a storage device such as a hard disk drive or a nonvolatile memory, and stores a sparse matrix 12a, a vector 12b, and a type 12c. The sparse matrix 12a includes a sparse matrix A of n rows × l columns and a sparse matrix B of n rows × k columns. Here, n is the codeword length.

The vector 12 b is a 1-row × l-column l-dimensional horizontal vector c ₂ with a Hamming weight (nP: where nP is a natural number). In the first embodiment, the case where an 1-dimensional horizontal vector having a Hamming weight of 2 is used will be described. However, an 1-dimensional horizontal vector in which the Hamming weight is an even weight other than 0 (for example, 4 or 6). It is good also as using.

  The type 12c includes a type P (a code word length n × type P is a natural number) taking a predetermined value. Note that the sparse matrix 12a, the vector 12b, and the type 12c are predetermined and take static values. In the following description, the sparse matrix 12a, the vector 12b, and the type 12c may be referred to as parameters.

The control unit 13 is a processing unit that performs overall control of various devices included in the encoding device 10. Moreover, the optimization execution unit 13a performs the storage unit 12 reads the parameters, the process passing from the message u ^k received from the input unit 11 to generate a codeword x ⁿ is a type constant code to the output unit 14 It is a processing unit.

  Hereinafter, the processing content performed by the optimization execution unit 13a will be described in more detail. Note that a or b used in the following description is an arbitrary binary number.

式（ａ１）および式（ａ２）が満足されるものとする。 As already described with reference to FIG. 1A, the communication path has a conditional probability W _{Y | X} for the output y when the input x is given. (Omitted). When the codeword length is n, the number of columns of the sparse matrix A is 1, the number of columns of the sparse matrix B is k, the entropy related to X is H (X), and the conditional entropy related to X is H (X | Y) ,

It is assumed that the formula (a1) and the formula (a2) are satisfied.

式（ａ３）で与えられる。 If P is a probability distribution on X and W _{Y | X} is a conditional probability, then entropy H (X) for X is

It is given by equation (a3).

式（ａ４）で与えられる。ただし、式（ａ４）は、式（ａ５）を満足するものとする。 Also, the conditional entropy H (X | Y) for X is

It is given by equation (a4). However, Formula (a4) shall satisfy Formula (a5).

式（１）としてあらわされる。なお、ｃ_２は、ハミング重みが２のｌ次元横ベクトル（ベクトル１２ｂ）であり、ｘ_ｉは、ｘ^ｎの第ｉ成分（ｉ桁目）である。 In this case, the encoder “Φ _n : {0, 1} ^k → {0, 1} ⁿ ” included in the optimization execution unit 13a assumes that an input message is u ^k ε {0, 1} ^k .

It is expressed as equation (1). Incidentally, _{c 2} is the Hamming weight of 2 l-dimensional horizontal vector (vector 12b), _{x i} is the i th component of ^{x n} (i-th digit).

Here, if Equation (1) can be regarded as a linear programming problem, an existing efficient calculation algorithm can be applied. However, since the matrix operation in each constraint condition of Expression (1) is an operation on the field F ₂ which is a binary field, it does not become a linear programming problem as it is.

Therefore, the optimization execution unit 13a converts each constraint condition of Expression (1) into a linear constraint condition using a relaxation method (see Non-Patent Document 4 described above), and a codeword x that satisfies these linear constraint conditions. ⁿ was determined using linear programming.

  Thereby, it is possible to suppress the calculation amount of the constant type code, which has been conventionally considered to increase exponentially with respect to the codeword length n, to the calculation amount of the polynomial order. Therefore, according to the encoding device 10 shown in FIG. 3, it is possible to realize encoding that efficiently generates a constant type code regardless of the length of the codeword.

  Next, a processing procedure executed by the encoding device 10 relating to the channel code will be described with reference to FIG. FIG. 4 is a flowchart showing a processing procedure executed by the encoding apparatus 10 relating to the channel code.

As shown in FIG. 4, when the message u ^k ε {0, 1} ^k is input to the input unit 11 (step S101), the optimization execution unit 13a reads each parameter from the storage unit 12 (step S102). .

Then, calculate the constraint condition of the equation (1), ^i.e., at ^{_{x n A = c 2, x}} n B = a ^{u k} and? X _i = all satisfied nP codeword ^{x n,} linear programming in combination with Relaxation (Step S103). Then, ^xn calculated in step S103 is passed to the output unit 14 as a code word (step S104), and the process is terminated.

  Next, a decoding device that decodes the codeword encoded by the encoding device 10 shown in FIG. 3 will be described. FIG. 5 is a block diagram showing a configuration of the decoding apparatus 20 related to the channel code.

  As illustrated in FIG. 5, the decoding device 20 includes an input unit 21, a storage unit 22, a control unit 23, and an output unit 24. The storage unit 22 stores a sparse matrix 22a, a vector 22b, and a type 22c, and the control unit 23 further includes an optimization execution unit 23a.

The input unit 21 is a device that receives a communication path output y ⁿ (y ⁿ ε {0, 1} ⁿ ) and performs processing for passing the received communication path output y ⁿ to the control unit 23. The output unit 24 receives the estimated message x ^{n B} from the control unit 23 is a device for performing the process of outputting the estimated message x ^{n B} received.

  The storage unit 22 is a storage unit configured by a storage device such as a hard disk drive or a nonvolatile memory, and stores a sparse matrix 22a, a vector 22b, and a type 22c. The sparse matrix 22a includes a sparse matrix A of n rows × l columns and a sparse matrix B of n rows × k columns. Here, n is the codeword length. Note that the sparse matrix A and the sparse matrix B are the same as the sparse matrix 12a stored in the storage unit 12 of the encoding device 10 illustrated in FIG.

The vector 22b is a 1-row × l-column l-dimensional horizontal vector c2 having a Hamming weight (nP: where nP is a natural number) of ₂ , and is stored in the storage unit 12 of the encoding device 10 shown in FIG. The same as the vector 12b to be executed. The type 22c is also the same as the type 12c stored in the storage unit 12 of the encoding device 10 illustrated in FIG. In the following description, the sparse matrix 22a, the vector 22b, and the type 22c may be referred to as parameters.

The control unit 23 is a processing unit that performs overall control of various devices included in the decoding device 20. Moreover, the optimization execution unit 23a reads out the parameters from the storage unit 22, performs to generate an estimated message x ^{n B} from the channel output y ⁿ received from the input unit 21 pass to the output section 24 the processing unit It is.

  Hereinafter, the processing content performed by the optimization execution unit 23a will be described in more detail. Note that a or b used in the following description is an arbitrary binary number.

式（２）としてあらわされる。なお、Ｈ（ｘ^ｎ｜ｙ^ｎ）は、系列ｘ^ｎおよび系列ｙ^ｎの同時経験確率分布から導かれる条件付エントロピーである。 When the decoder “Ψ _n : {0, 1} ⁿ → {0, 1} ^k ” included in the optimization execution unit 23 a is y ⁿ ε {0, 1} ⁿ ,

It is expressed as equation (2). ^{Incidentally, H (x n | y n} ) is the conditional entropy derived from co empirical probability distribution of the sequence ^{x n} and sequence ^{y n.}

式（ａ６）は、式（ａ７）を達成するｘをあらわす。 Also,

Equation (a6) represents x that achieves equation (a7).

Here, if Equation (2) can be regarded as a linear programming problem, an existing efficient calculation algorithm can be applied. However, since the matrix operation in each constraint condition of Expression (2) is an operation on the field F ₂ which is a binary field, it does not become a linear programming problem as it is. Furthermore, the objective function of equation (2) is not linear with respect to the variable x _i .

  Therefore, the optimization execution unit 23a converts each constraint condition of Expression (2) into a linear constraint condition using a relaxation method (see Non-Patent Document 4 described above). Further, for the objective function of Equation (2), the following method for obtaining an executable solution is used.

式（３）とした場合、式（２）に示した復号化器Ψ_ｎを、

式（４）へ置き換えることとした。 Specifically, the communication channel output is represented by y ⁿ , N (a | y ⁿ ) = {number of occurrences of a in sequence y ⁿ }, and natural number m.

In the case of Equation (3), the decoder Ψ _n shown in Equation (2) is

It was decided to replace the equation (4).

However, it is _Hb ((delta)) =-deltalogdelta- (1-delta) log (1-delta). X _i is the i-th component (i-th digit) of x ⁿ , and y _i is the i-th component (i-th digit) of y ⁿ .

式（４’）を最小にするｘ^ｎを求めるものである。 That is, equation (4) is an objective function under linear constraints.

^Xn that minimizes the expression (4 ′) is obtained.

Then, the optimization execution unit 23a determines whether or not there is an executable solution of Expression (4) for each m, and if there is an executable solution, repeats the procedure of calculating Expression (4 ′). To finally obtain x ⁿ giving the minimum value of the equation (4 ′).

  Here, it should be noted that the above-described procedure can be executed entirely by applying linear programming. That is, it is possible to suppress the calculation amount of the constant type code, which is conventionally considered to increase exponentially with respect to the codeword length n, to the calculation amount of the polynomial order. Thus, the present invention also newly proposes an algorithm for realizing optimization when conditional entropy is an objective function.

  Next, a processing procedure executed by the decoding device 20 regarding the channel code will be described with reference to FIG. FIG. 6 is a flowchart showing a processing procedure executed by the decoding apparatus 20 regarding the channel code.

As shown in FIG. 6, when the channel output y ⁿ ε {0, 1} ⁿ is input to the input unit 21 (step S201), the optimization execution unit 23a reads each parameter from the storage unit 22 (step S201). S202).

Then, each constraint condition of Expression (4), that is, x ⁿ A = c ₂ , Σx _i = nP and Σy _i x _i = m are all satisfied, and the objective function (see Expression (4 ′)) is minimized. ^Xn to be calculated is calculated by linear programming using a relaxation method together (step S203). Then, x ⁿ B, which is the product of x ⁿ calculated in step S203 and sparse matrix B, is passed to the output unit 24 as an estimation message (step S204), and the process is terminated.

As described above, in the first embodiment, the storage unit stores the first sparse matrix, the second sparse matrix, the one-row horizontal matrix having a predetermined Hamming weight, and the predetermined type, and the optimization execution unit Is the first constraint equation that the product of the codeword and the first sparse matrix is equal to the row matrix of one row having a predetermined Hamming weight, and the product of the codeword and the second sparse matrix is The second constraint expression that is equal to the binary message, and the Hamming weight that is the product of the codeword length and the type of the codeword is equal to the number of digits in which the codeword element is 1 The first constraint equation, the second constraint equation, and the third constraint equation are converted into linear constraint conditions on the real field by a relaxation method, and the linear constraint conditions are The encoder is configured to solve all the codewords that satisfy all by linear programming. Therefore, it is possible to realize encoding that efficiently generates a constant type code regardless of the length of the codeword. In addition, a transmission capacity closer to the Shannon limit can be realized than a code using a typical sequence.

Further, in the first embodiment, the storage unit stores the first sparse matrix, the second sparse matrix, the row matrix having one row having a predetermined Hamming weight, and a predetermined type, which are used for encoding. A first constraint expression, code, in which the execution unit assumes that the product of the code word representing the encoded binary message and the first sparse matrix is equal to a one-row row matrix having a predetermined Hamming weight A second constraint expression in which the Hamming weight, which is the product of the codeword length of the word and the type, is equal to the number of digits whose codeword element is 1, the codeword element and the channel output element; Using a linear objective function with respect to the codeword, the first constraint equation, the second constraint equation, and the third constraint equation where the number of digits in which both are 1 is equal to the predetermined natural number After converting the third constraint equation to a linear constraint on the real number field using the relaxation method, A codeword that minimizes an objective function among codewords that satisfy all the conditions is solved by linear programming, and a decoding device is set so that a product of the obtained codeword and the second sparse matrix is a binary message. Configured. Therefore, a constant type code can be efficiently decoded.

  In a second embodiment described below, a case where the present invention is applied to a distorted information source code will be described.

  In the second embodiment, a case where the present invention is applied to a distorted information source code will be described. First, the configuration of the encoding device will be described with reference to FIG. FIG. 7 is a block diagram illustrating a configuration of the encoding device 30 related to the distorted information source code.

  As illustrated in FIG. 3, the encoding device 30 includes an input unit 31, a storage unit 32, a control unit 33, and an output unit 34. The storage unit 32 stores a sparse matrix 32a, a vector 32b, and a type 32c, and the control unit 33 further includes an optimization execution unit 33a.

The input unit 31 is a device that receives an information source output x ⁿ (x ⁿ ε {0, 1} ⁿ ) to be encoded and performs a process of passing the received information source output x ⁿ to the control unit 33. The output unit 34 is a device that receives the codeword y ⁿ B from the control unit 33 and performs processing to output the received codeword y ⁿ B.

  The storage unit 32 is a storage unit configured by a storage device such as a hard disk drive or a nonvolatile memory, and stores a sparse matrix 32a, a vector 32b, and a type 32c. The sparse matrix 32a includes a sparse matrix A of n rows × l columns and a sparse matrix B of n rows × k columns. Here, n is the codeword length.

Also, vector 32b is Hamming weight (nP: However, nP is a natural number) is a l-dimensional horizontal vector _{c 2} of 2 of one row × l column. In the second embodiment, a case where an l-dimensional horizontal vector having a hamming weight of 2 is used will be described. However, an l-dimensional horizontal vector having a hamming weight other than 0 (eg, 4 or 6). It is good also as using.

  The type 32c includes a type P (a code word length n × type P is a natural number) taking a predetermined value. The sparse matrix 32a, the vector 32b, and the type 32c are predetermined and take static values. In the following description, the sparse matrix 32a, the vector 32b, and the type 32c may be referred to as parameters.

The control unit 33 is a processing unit that performs overall control of various devices included in the encoding device 30. Further, the optimization execution unit 33a reads out each parameter from the storage unit 32, generates a decoder output y ⁿ that is a constant type code from the information source output x ⁿ received from the input unit 31, and generates the generated y ⁿ Is a processing unit that performs a process of passing a codeword y ⁿ B, which is a product of sparse matrix B, to the output unit 34.

  Hereinafter, the processing content performed by the optimization execution unit 33a will be described in more detail. Note that a or b used in the following description is an arbitrary binary number.

As already described in FIG. 1B, the information source is represented by the probability P _x . Further, the encoder [Phi _n has received the information source output x ⁿ from the information source, the information source output x ⁿ to perform compression operations by encoding the u ^k.

式（ｂ１）を満足することが要請される。 Then, if the distortion measure is “d: {0, 1} × {0, 1} → [0, ∞]”, the information source output x ⁿ and the decoder output y ⁿ are given by the distortion criterion D given in advance. (D> 0)

It is required to satisfy formula (b1).

式（ｂ２）および式（ｂ３）が満足されるものとする。 If the number of columns of the sparse matrix A is 1, the number of columns of the sparse matrix B is k, the entropy related to Y is H (Y), and the conditional entropy related to Y is H (Y | X),

It is assumed that the formulas (b2) and (b3) are satisfied.

Here, the entropy H (Y) related to Y and the conditional entropy H (Y | X) related to Y are respectively Px as a probability distribution on X, a predetermined conditional probability (for example, a condition for achieving a rate / distortion limit) The entropy related to Y and the conditional entropy when the attached isolation value is W _{Y | X.} Then, the conditional probability W _{Y | X} that achieves the rate / distortion limit can be obtained using, for example, the Arimoto-Blahut algorithm (see Non-Patent Document 1).

式（５）としてあらわされる。なお、ｃ_２は、ハミング重みが２のｌ次元横ベクトル（ベクトル３２ｂ）である。 In this case, the encoder “Φ _n : {0, 1} ⁿ → {0, 1} ^k ” included in the optimization execution unit 33a is configured to output the information source output x ⁿ ε {0, 1} ⁿ .

This is expressed as equation (5). Incidentally, _{c 2} is a Hamming weight of 2 l-dimensional horizontal vector (vector 32 b).

式（ｂ４）、式（ｂ５）および式（ｂ６）が満たされるものとする。なお、Ｎ（ａ，ｂ｜ｘ^ｎｙ^ｎ）＝｛系列ｘ^ｎｙ^ｎにおけるａおよびｂの同時出現回数｝であり、式（ｂ６）は、条件付ダイバージェンスである。 here,

It is assumed that Expression (b4), Expression (b5), and Expression (b6) are satisfied. Note that N (a, b | x ⁿ y ⁿ ) = {number of simultaneous occurrences of a and b in the sequence x ⁿ y ⁿ }, and equation (b6) is conditional divergence.

Here, if Expression (5) can be regarded as a linear programming problem, an existing efficient calculation algorithm can be applied. However, since the matrix operation in each constraint condition of Equation (5) is an operation on the field F ₂ that is a binary field, it does not become a linear programming problem as it is. Furthermore, the objective function of equation (5) is not linear with respect to the variable y _i .

  Therefore, the optimization execution unit 33a converts each constraint condition of Expression (5) into a linear constraint condition using a relaxation method (see Non-Patent Document 4 described above). Further, for the objective function of Equation (5), the following method for obtaining an executable solution is used.

式（７）とした場合、式（５）に示した符号化器Φ_ｎを、

式（８）へ置き換えることとした。 Specifically, the information source output is x ⁿ and the natural number m is

In the case of formula (7), the encoder Φ _n shown in formula (5) is

It was decided to replace the equation (8).

式（ｂ７）を満足するものとする。 However, for p (0 <p <1) and q (0 <q <1),

It is assumed that the formula (b7) is satisfied.

式（８’）を最小にするｙ^ｎを求めるものである。 That is, equation (8) is an objective function under linear constraints.

Equation (8 ') is intended to determine the y ⁿ be minimized.

Then, the optimization execution unit 33a determines whether or not there is an executable solution of Expression (8) for each m, and if there is an executable solution, repeats the procedure of calculating Expression (8 ′). by ultimately determine the y ⁿ giving the minimum value of formula (8 ').

  Here, it should be noted that the above-described procedure can be executed entirely by applying linear programming. That is, it is possible to suppress the calculation amount of the constant type code, which is conventionally considered to increase exponentially with respect to the codeword length n, to the calculation amount of the polynomial order. Thus, the present invention also newly proposes an optimization implementation algorithm in the case where conditional divergence is an objective function.

  Next, a processing procedure executed by the encoding device 30 relating to the distorted information source code will be described with reference to FIG. FIG. 8 is a flowchart showing a processing procedure executed by the encoding device 30 regarding the distorted information source code.

As shown in FIG. 8, when the information source output x ⁿ ε {0, 1} ⁿ is input to the input unit 31 (step S301), the optimization execution unit 33a reads each parameter from the storage unit 32 (step S301). S302).

Then, each constraint condition of Expression (8), that is, y ⁿ A = c ₂ , Σy _i = nP, and Σx _i y _i = m are all satisfied, and the objective function (see Expression (8 ′)) is minimized. the ^{y n} which is calculated by the linear programming combination with a relaxation method (step S303). Then, step S303 is the product of the calculated ^{y n} and sparse matrix B by passing a ^{y n} B to the output section 34 as a code word (step S304), and ends the process.

  Next, a decoding apparatus that decodes the codeword encoded by the encoding apparatus 30 shown in FIG. 7 will be described. FIG. 9 is a block diagram showing the configuration of the decoding apparatus 40 related to the distorted information source code.

  As shown in FIG. 9, the decryption device 40 includes an input unit 41, a storage unit 42, a control unit 43, and an output unit 44. The storage unit 42 stores a sparse matrix 42a, a vector 42b, and a type 42c, and the control unit 43 further includes an optimization execution unit 43a.

The input unit 41 is a device that receives a codeword u ^k (u ^k ε {0, 1} ^k ) and performs a process of passing the received code word u ^k to the control unit 43. The output unit 44 receives the restoration message y ⁿ from the control unit 43 is a device for performing the process of outputting the restored message y ⁿ received.

  The storage unit 42 is a storage unit configured by a storage device such as a hard disk drive or a non-volatile memory, and stores a sparse matrix 42a, a vector 42b, and a type 42c. The sparse matrix 42a includes a sparse matrix A of n rows × l columns and a sparse matrix B of n rows × k columns. Here, n is the codeword length. Note that the sparse matrix A and the sparse matrix B are the same as the sparse matrix 32a stored in the storage unit 32 of the encoding device 30 illustrated in FIG.

The vector 42b is a 1-row × l-column l-dimensional horizontal vector c2 having a Hamming weight (nP: where nP is a natural number) of ₂ , and is stored in the storage unit 32 of the encoding device 30 shown in FIG. It is the same as the vector 32b to be executed. The type 42c is the same as the type 32c stored in the storage unit 32 of the encoding device 30 shown in FIG. In the following description, the sparse matrix 42a, the vector 42b, and the type 42c may be referred to as parameters.

The control unit 43 is a processing unit that performs overall control of various devices included in the decoding device 40. Moreover, the optimization execution unit 43a reads out the parameters from the storage unit 42 is the processing unit that performs processing to pass thereby generating a restoration message y ⁿ from the code word u ^k received from the input unit 41 to the output section 44 .

  Hereinafter, the processing content performed by the optimization execution unit 43a will be described in more detail. Note that a or b used in the following description is an arbitrary binary number.

式（６）としてあらわされる。なお、ｃ_２は、ハミング重みが２のｌ次元横ベクトル（ベクトル４２ｂ）であり、ｙ_ｉは、ｙ^ｎの第ｉ成分（ｉ桁目）である。 The decoder “Ψ _n : {0, 1} ^k → {0, 1} ⁿ ” included in the optimization execution unit 43 a converts the code word encoded by the encoding device 30 into u ^k ε {0, 1}. ^{If k} ,

It is expressed as equation (6). Incidentally, _{c 2} is the Hamming weight of 2 l-dimensional horizontal vector (vector 42b), _{y i} is the i-th component of ^{y n} (i-th digit).

Here, if Equation (6) can be regarded as a linear programming problem, an existing efficient calculation algorithm can be applied. However, since the matrix operation in each constraint condition of Expression (6) is an operation on the field F ₂ which is a binary field, it does not become a linear programming problem as it is.

Therefore, the optimization execution unit 43a converts each constraint condition of Expression (6) into a linear constraint condition using a relaxation method (see Non-Patent Document 4 described above), and a codeword y that satisfies these linear constraint conditions. ⁿ was determined using linear programming.

  Thereby, it is possible to suppress the calculation amount of the constant type code, which has been conventionally considered to increase exponentially with respect to the codeword length n, to the calculation amount of the polynomial order. Therefore, according to the encoding device 40 shown in FIG. 9, it is possible to efficiently decode a constant type code regardless of the length of the codeword.

  Next, a processing procedure executed by the decoding apparatus 40 regarding the distorted information source code will be described with reference to FIG. FIG. 10 is a flowchart showing a processing procedure executed by the decoding apparatus 40 regarding the distorted information source code.

As shown in FIG. 10, when the code word u ^k ε {0, 1} ^k is input to the input unit 41 (step S401), the optimization execution unit 43a reads each parameter from the storage unit 42 (step S402). ).

Then, calculate the constraint condition of the equation (6), ^i.e., in ^{_{y n A = c 2, y}} n B = a ^{u k} and .sigma.y _i = all satisfied nP restoration message ^{y n,} linear programming in combination with Relaxation (Step S403). Then, passing a ^{y n} calculated in step S403 as the recovered message to the output unit 44 (step S404), and ends the process.

As described above, in the second embodiment, the storage unit stores the first sparse matrix, the second sparse matrix, the one-row horizontal matrix having a predetermined Hamming weight, and the predetermined type, and the optimization execution unit. but first constraint, the code word length decoder output and the type of the product of the decoder output and a first sparse matrix is a transverse matrix and equal values of a line having a predetermined Hamming weight The second constraint that the Hamming weight, which is the product of the two, is equal to the number of digits of the decoder output element equal to 1, both the source output element and the decoder output element are both A third constraint expression that the number of digits that are 1 is equal to a predetermined natural number, and a linear objective function with respect to the decoder output, the first constraint expression, the second constraint expression, and the second constraint expression are used. After converting the constraint equation (3) into linear constraints on the real field by the relaxation method, Among the decoder outputs satisfying the above, the decoder output having the minimum objective function is obtained by linear programming, and the product of the obtained decoder output and the second sparse matrix is used as a distorted information source code. Thus, the encoding apparatus was configured. Therefore, it is possible to realize encoding that efficiently generates a constant type code regardless of the length of the codeword.

In the second embodiment, the storage unit stores the first sparse matrix, the second sparse matrix, the row matrix having a predetermined hamming weight, and a predetermined type, which are used for encoding. The execution unit assumes that the product of the decoder output and the first sparse matrix is equal to the one row row matrix having a predetermined Hamming weight, the first constraint equation, the decoder output, and the second The second constraint equation that the product with the sparse matrix is equal to the distortion information source code, and the Hamming weight that is the product of the codeword length and the type of the decoder output is an element of the decoder output Using a third constraint expression that is equal to the number of digits for which 1 is 1, and using the relaxation method, the first constraint equation, the second constraint equation, and the third constraint equation are linear constraint conditions on the real number field. The decoder so that the decoder output satisfying all these linear constraints is solved by linear programming. Configuration was. Therefore, a constant type code can be efficiently decoded.

  By the way, among the processes described in the above-described embodiments, all or a part of the processes described as being automatically performed can be manually performed, or the processes described as being performed manually can be performed. All or a part can be automatically performed by a known method.

  Further, each component of each illustrated apparatus is functionally conceptual, and does not necessarily need to be physically configured as illustrated. In other words, the specific form of distribution / integration of each device is not limited to that shown in the figure, and all or a part thereof may be functionally or physically distributed or arbitrarily distributed in arbitrary units according to various loads or usage conditions. Can be integrated and configured.

  Furthermore, each processing function performed by each device is entirely or arbitrarily partly performed by a computer (for example, a personal computer or a workstation) and a program that is analyzed and executed by the computer, or by hardware using wired logic. Can be realized as

  Such a program can be distributed through a network such as the Internet. Further, such a program can be executed by being recorded on a computer-readable recording medium such as a hard disk, a flexible disk (FD), a CD-ROM, an MO, and a DVD, and being read from the recording medium by the computer.

  As described above, the encoding device, the decoding device, the encoding method, the decoding method, the encoding program, and the decoding program according to the present invention efficiently generate a constant type code regardless of the length of the codeword. It is useful when it is desired to perform encoding and decoding thereof, and is particularly suitable for implementation in various technologies using a fixed type code, such as radio system technology and search technology.

DESCRIPTION OF SYMBOLS 10 Coding device 11 Input unit 12 Storage unit 12a Sparse matrix 12b Vector 12c type 13 Control unit 13a Optimization execution unit 14 Output unit 20 Decoding device 21 Input unit 22 Storage unit 22a Sparse matrix 22b Vector 22c type 23 Control unit 23a Optimal Encoding unit 24 Output unit 30 Encoding device 31 Input unit 32 Storage unit 32a Sparse matrix 32b Vector 32c Type 33 Control unit 33a Optimization execution unit 34 Output unit 40 Decoding device 41 Input unit 42 Storage unit 42a Sparse matrix 42b Vector 42c Type 43 Control unit 43a Optimization execution unit 44 Output unit

Claims (16)

An encoding device that generates a channel code having a constant Hamming weight from a binary message,
Storage means for storing a first sparse matrix, a second sparse matrix, a row of rows having a predetermined Hamming weight, and a predetermined type;
A first constraint expression that a product of a codeword and the first sparse matrix is equivalent to a row matrix having a predetermined Hamming weight, and the codeword and the second sparse matrix A second constraint expression in which a product is equivalent to the binary message, and a digit in which the Hamming weight, which is a product of the codeword length of the codeword and the type, has 1 in the codeword The first constraint equation, the second constraint equation, and the third constraint equation are converted into linear constraint conditions on a real field by a relaxation method using a third constraint equation that is equal to the number of And an optimization execution unit that solves the codeword satisfying all of the linear constraint conditions by linear programming. A decoding device that generates a binary message before encoding from a channel output,
Storage means for storing a first sparse matrix, a second sparse matrix, a row matrix having a predetermined Hamming weight, and a predetermined type used for encoding;
A first constraint equation, wherein the product of the encoded code word representing the binary message and the first sparse matrix is equal to the one-row horizontal matrix having the predetermined Hamming weight , the code A second constraint equation that assumes that the Hamming weight, which is the product of the codeword length of the word and the type, is equal to the number of digits in which the codeword element is 1, the codeword element and the communication A third constraint expression in which the number of digits whose path output elements are both 1 is equal to a predetermined natural number, and the first constraint expression using a linear objective function with respect to the codeword The second constraint equation and the third constraint equation are converted into linear constraint conditions on a real field by a relaxation method, and the objective function is minimized among the codewords that satisfy all the linear constraint conditions. The codeword to be solved by linear programming, and the codeword and the first Decoding apparatus characterized by comprising an optimizing execution means to said binary message the product of the sparse matrix. The objective function used by the optimization execution unit is:
The code word is x ⁿ , the communication channel output is y ⁿ , the code word length is n, the type is P, and Hb (δ) is H _b (δ) = − δ log δ− (1-δ) log (1− δ), when the number of occurrences of a in the sequence r ⁿ is N (a | r ⁿ ), and the predetermined natural number m is 0 ≦ m ≦ min {nP, N (1 | y ⁿ )},
(N (0 | y ⁿ ) / n) H _b ((nP−m) / N (0 | y ⁿ )) +
(N (1 | y ⁿ ) / n) H _b (m / N (1 | y ⁿ ))
The decoding device according to claim 2, wherein The predetermined natural number is
0 or more and less than or equal to the smaller value of the number of digits in which the Hamming weight or the channel output element is 1.
The optimization execution means includes:
For each of the predetermined natural numbers, determine whether there is an executable solution for the codeword that satisfies all of the first constraint equation, the second constraint equation, and the third constraint equation, and 4. The decoding apparatus according to claim 2, wherein the value of the objective function is calculated when it is determined that there is an executable solution. An encoding device that generates a distorted information source code having a constant Hamming weight based on a decoder output obtained from an information source output,
Storage means for storing a first sparse matrix, a second sparse matrix, a row of rows having a predetermined Hamming weight, and a predetermined type;
First constraint that the product of the said decoder outputs the first sparse matrix is a transverse matrix and equal values of a line having a predetermined Hamming weight, the codeword length of the decoder output The Hamming weight, which is the product of the type and the type, is equal to the number of digits in which the element of the decoder output is 1, the second constraint equation, the element of the information source output and the decoder Using the third constraint equation that the number of digits whose output elements are both 1 is equal to a predetermined natural number and a linear objective function with respect to the decoder output, the first constraint And converting the equation, the second constraint equation, and the third constraint equation into linear constraints on a real number field by a relaxation method, and the objective function among the decoder outputs satisfying all the linear constraints is The decoder output to be minimized is solved by linear programming, and the decoder output Encoding apparatus is characterized in that an optimization execution means for said second product of sparse matrix and said lossy encoded information with. The objective function used by the optimization execution unit is:
The information source output is x ⁿ , the decoder output is y ⁿ , the codeword length is n, the type is P, p is 0 <p <1, q is 0 <q <1, D (p‖q ) and D (p‖q) = plog (p / q) + (1-p) log ((1-p) / (1-q)), the number of occurrences of a in the sequence ^{r n} n (a | ^{r n} ), where the predetermined conditional probability is W _{Y | X}
(N (0 | y ⁿ ) / n) D ((nP−m) / N (0 | x ⁿ ) ‖W _{Y | X} (1 | 0)) +
(N (1 | y ⁿ ) / n) D (m / N (1 | y ⁿ ) ‖W _{Y | X} (1 | 1))
The encoding apparatus according to claim 5, wherein The predetermined natural number is
0 or more and less than or equal to the smaller value of the number of digits in which the Hamming weight or the information source output element is 1.
The optimization execution means includes:
For each of the predetermined natural numbers, a determination is made as to whether there is an executable solution for the decoder output that satisfies all of the first constraint equation, the second constraint equation, and the third constraint equation. The encoding apparatus according to claim 5, wherein the value of the objective function is calculated when it is determined that the feasible solution exists. A decoding device for generating a decoder output from a distorted information source code,
Storage means for storing a first sparse matrix, a second sparse matrix, a row matrix having a predetermined Hamming weight, and a predetermined type used for encoding;
A first constraint expression that a product of the decoder output and the first sparse matrix is equal to a row matrix having a predetermined Hamming weight , the decoder output, and the second The sparse matrix product is equal to the distorted information source code, and the Hamming weight is a product of the codeword length of the decoder output and the type. Using the third constraint equation that assumes that the number of digits of the decoder output is equal to 1, the first constraint equation, the second constraint equation, and the third constraint equation are relaxed A decoding apparatus comprising: optimization executing means for converting the output of the decoder that satisfies all of the linear constraint conditions by linear programming after conversion to a linear constraint condition on a real field by a method . An encoding method for generating a channel code having a constant Hamming weight from a binary message,
A storage step of storing a first sparse matrix, a second sparse matrix, a one-row row matrix having a predetermined Hamming weight, and a predetermined type in a storage unit;
A first constraint expression that a product of a codeword and the first sparse matrix is equivalent to a row matrix having a predetermined Hamming weight, and the codeword and the second sparse matrix A second constraint expression in which a product is equivalent to the binary message, and a digit in which the Hamming weight, which is a product of the codeword length of the codeword and the type, has 1 in the codeword The first constraint equation, the second constraint equation, and the third constraint equation are converted into linear constraint conditions on a real field by a relaxation method using a third constraint equation that is equal to the number of And an optimization execution step of solving the codeword satisfying all of the linear constraint conditions by linear programming. A decoding method for generating a binary message before encoding from a channel output,
A storage step of storing a first sparse matrix, a second sparse matrix, a row matrix having a predetermined Hamming weight, and a predetermined type used for encoding in a storage unit;
A first constraint equation, wherein the product of the encoded code word representing the binary message and the first sparse matrix is equal to the one-row horizontal matrix having the predetermined Hamming weight , the code A second constraint equation that assumes that the Hamming weight, which is the product of the codeword length of the word and the type, is equal to the number of digits in which the codeword element is 1, the codeword element and the communication A third constraint expression in which the number of digits whose path output elements are both 1 is equal to a predetermined natural number, and the first constraint expression using a linear objective function with respect to the codeword The second constraint equation and the third constraint equation are converted into linear constraint conditions on a real field by a relaxation method, and the objective function is minimized among the codewords that satisfy all the linear constraint conditions. The codeword to be solved by linear programming, and the codeword and the first Decoding method comprising the product of the sparse matrix of the containing and optimizing execution step to the binary message. An encoding method for generating a distorted information source code having a constant Hamming weight based on a decoder output obtained from an information source output,
A storage step of storing a first sparse matrix, a second sparse matrix, a one-row row matrix having a predetermined Hamming weight, and a predetermined type in a storage unit;
First constraint that the product of the said decoder outputs the first sparse matrix is a transverse matrix and equal values of a line having a predetermined Hamming weight, the codeword length of the decoder output The Hamming weight, which is the product of the type and the type, is equal to the number of digits in which the element of the decoder output is 1, the second constraint equation, the element of the information source output and the decoder Using the third constraint equation that the number of digits whose output elements are both 1 is equal to a predetermined natural number and a linear objective function with respect to the decoder output, the first constraint Of the decoder output satisfying all of the linear constraint conditions after converting the formula, the second constraint formula, and the third constraint formula into linear constraint conditions on a real number field by a relaxation method And solves the decoder output by minimizing the decoder by linear programming. Coding method characterized in that the product of force and said second sparse containing and optimizing execution step of said lossy encoded information. A decoding method for generating a decoder output from a distorted information source code, comprising:
A storage step of storing a first sparse matrix, a second sparse matrix, a row matrix having a predetermined Hamming weight, and a predetermined type used for encoding in a storage unit;
A first constraint expression that a product of the decoder output and the first sparse matrix is equal to a row matrix having a predetermined Hamming weight , the decoder output, and the second The sparse matrix product is equal to the distorted information source code, and the Hamming weight is a product of the codeword length of the decoder output and the type. Using the third constraint equation that assumes that the number of digits of the decoder output is equal to 1, the first constraint equation, the second constraint equation, and the third constraint equation are relaxed And an optimization execution step of solving the decoder output satisfying all of the linear constraint conditions by linear programming after conversion to a linear constraint condition on a real number field by a method .   An encoding program for causing a computer to execute an operation procedure of the encoding apparatus according to claim 1.   A decoding program that causes a computer to execute the operation procedure of the decoding device according to claim 2, 3 or 4.   An encoding program for causing a computer to execute an operation procedure of the encoding apparatus according to claim 5, 6 or 7.   A decoding program that causes a computer to execute the operation procedure of the decoding device according to claim 8.

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