CN113824451B - Construction method of QC-LDPC code based on Jacobsthal array - Google Patents

Abstract

A construction method of QC-LDPC codes based on Jacobsthal arrays comprises the following steps: constructing a QC-LDPC code of a dual diagonal basis matrix structure by using Jacobsthal arrays; the design can generate a short-loop elimination method with large girth, and the large girth QC-LDPC code based on Jacobsthal number series is constructed by modifying the cyclic permutation matrix coefficient. The invention designs the QC-LDPC code based on the Jacobsthal array, which realizes the quick coding and the improvement of the coding performance compared with the F-QC-LDPC code and has better feasibility. A QC-LDPC code based on a Jacobsthal-Lucas sequence is designed, and qualitative explanation is made on the performance stability of the code. The invention directly eliminates the short ring of the JACO-QC-LDPC code and the JACO-LUCAS-QC-LDPC code after mask modification, and improves the coding performance of flexible code groups with any code length at one time. An algorithm for counting and eliminating 4-ring and 6-ring is designed, and a JACO-QC-LDPC code with large girth is successfully constructed.

Description

Construction method of QC-LDPC code based on Jacobsthal array

Technical field:

the invention relates to the technical field of digital communication, in particular to a construction method of QC-LDPC codes based on Jacobsthal arrays.

The background technology is as follows:

channel coding is the cornerstone of all modern communication systems. For decades, the channel coding technology approaches shannon limit continuously, which promotes the development of human communication technology, and achieves one more refulgence result.

The LDPC code is a block code based on a sparse parity check matrix, which is proposed in the doctor paper in 1963 by Robert Gallager of the university of Massa Medicata Fermentata, is almost suitable for any channel, belongs to a composite code like the Turbo code, has close performance, but is simpler and more convenient in decoding than the Turbo code.

The LDPC codes are classified into a plurality of categories, and after analysis through different angles and different characteristics, the LDPC codes can be distinguished into: regular LDPC codes and irregular LDPC codes; dividing LDPC code into: binary LDPC codes (binary LDPC) and high-order finite field GF (q) LDPC codes (non-binary LDPC codes) satisfy q=2 ^m The method comprises the steps of carrying out a first treatment on the surface of the The LDPC code can be distinguished into: LDPC block codes, generalized LDPC codes, and LDPC convolutional codes. The long code coding scheme of the 5G data channel coding in the Risbee conference voting is determined by using the LDPC code, and the coding mode of the short code is selected for reclassification. In the Athens conference, the 3GPP finally decides that the control channel coding scheme adopts Polar codes, and the data channel short code scheme still adopts LDPC codes.

The invention comprises the following steps:

in view of this, it is necessary to provide a construction method of QC-LDPC codes based on Jacobsthal arrays.

A construction method of QC-LDPC codes based on Jacobsthal arrays comprises the following steps:

constructing a QC-LDPC code of a dual diagonal basis matrix structure by using Jacobsthal arrays; the design can generate a short-loop elimination method with large girth, and the large girth QC-LDPC code based on Jacobsthal number series is constructed by modifying the cyclic permutation matrix coefficient.

Preferably, the Jacobsthal array { J _n Defined as:

J _n ＝J _n-1 +2J _n-2 (1-1)

in the formula (1-1) { J _n First two items J ₀ And J ₁ The formula II is as follows:

the recurrence formula for the Fibonacci array F (n), also known as the golden section array, is:

F(0)＝0 F(1)＝1 (1-3)

F(n)＝F(n-1)+F(n-2) (n≥2,n∈N ^* ) (1-4)

formulae (1-4) and (1-5) can be successively deduced { J } _n The } antecedents are respectively: 0. 1, 3, 5, 11, … …, however, if one needs to be found directly, the first few specific values need to be known in advance, which is useful for determining { J ] _n The term value of the specified term of the sequence is quite inconvenient, so that a common generating function phi (t) of the sequence needs to be determined to obtain a recurrence relation, wherein the expression of the recurrence generating function phi (t) is shown as the following formula (1-6):

when the formula (1-7) is satisfied:

then for any real number c, it satisfies the equation (1-8) as:

let Jacobsthal number column { J _n The occurrence function Φ (t) satisfying the equation (1-11) is:

then the formula (1-12) is:

the final formulae (1-13) and (1-14) are:

the base matrix structure is a double diagonal structure, a base matrix of the QC-LDPC code is constructed by using Jacobsthal number columns, and the base matrix adopts the double diagonal structure and has the expression:

H _b ＝[H _v |H _c ] (1-15)

wherein H is _c Adopts a quasi-dual diagonal structure in IEEE 802.16e, H _c The expression of (2) is the formula (1-16), wherein h is satisfied _c (1)＝h _c (m _b ) The value range of r satisfies that r is more than or equal to 2 and less than or equal to m _b -1；

In the construction of the base matrix, H _c The sizes of (2) are 5×5, let h _c (1)＝h _c (m _b )＝7，h _c (r) =0, and left submatrix H _v All elements of the first row of (a) are configured as "0", the first column is the head of the rest of the rows except the elements of the first row, the elements of the first column are the digits of the even number of the Jacobsthal number column, such as: 0. 1, 5, 21, … …, left subarray H _v Each row of numbers is a continuous Jacobsthal number column, H _v Except for the first row elements, and sequentially starting to sort backwards, thereby completing the QC-LDPC code left subarray H based on Jacobsthal array _v And finally obtaining the base matrix H of the QC-LDPC code based on Jacobsthal array _b Wherein H is _v The expression of (2) is shown as the expression (1-17), when the code rate is 1/2, H _v The size is 5 multiplied by 5; when the code rate is 2/3, H _v The size is 5 multiplied by 10;

preferably, the QC-LDPC code based on the Jacobsthal array structure is called as a JACO-QC-LDPC code, and the JACO-QC-LDPC code eliminates a short ring, and the specific method is as follows: traversing according to a 4-ring method, and designing a unit cyclic permutation matrix

The cyclic permutation matrix coefficient a is known _i,j The cyclic permutation matrix has a size p×p, p is the spreading factor, and the non-zero element, i.e., element "1", in the cyclic permutation matrix is located at a position (0, a) _i,j )、(1,a _i,j +1)、(2,a _i,j +2)、…、(p-a _i,j ,p)、(p-a _i,j +1,1)、...、(p-1,a _i,j -1) wherein the cyclic permutation matrix +.>

As a base matrix H _b Is expressed by the following formula

And satisfies i=1, 2, …, m; j=1, 2, …, n, as can be seen from the above, when a _i,j Not equal to-1, i.e. the cyclic permutation matrix

For a non-zero matrix, <' > a matrix of zero>

The non-zero element, element "1", is present in the position (x, mod [ (x+a) _i,j ),p]) Where x=0, 1, …, p-1 is satisfied;

setting information forming the existence of the 4-ring and circulating replacement matrix where check nodes are located

Corresponding coefficients of (a) are respectively

Assuming that the positions of the information nodes and the check nodes constituting the 4-ring are (m ₁ ,n ₁ )、(m ₂ ,n ₂ )、(m ₃ ,n ₃ )、(m ₄ ,n ₄ ) From the element positions of the respective information nodes and check nodes, it is possible and prescribed according to the formulas (1-18):

according to

The non-zero element, element "1", is present in the position (x, mod [ (x+a) _i,j ),p]) The method can obtain:

the formulae (1-19) and (1-20) are jointly obtainable:

finally, the formula (1-22) is simplified to obtain:

due to the base matrix H _b Smaller, so that the base matrix H can be directly modified by the masking technology according to the specific shape of the 4-ring _b Traversing by adopting a traversing search 4-ring method, and obtaining 4 cyclic permutation matrix coefficients a by single traversing _i,j Substituting the four cyclic permutation matrices into the expression (1-22) for verification, and if the expression (1-22) is satisfied, recognizing the four cyclic permutation matrices

Each information node is provided with 1 information node to form a 4-ring, the searched 4-rings are stored, and then the key cyclic permutation matrix coefficient a is carried out _ij Setting to-1 to complete the +.>

The 4 rings existing in the inner part are eliminated, so that the purpose of eliminating the rings is achieved.

Preferably, the JACO-QC-LDPC code is eliminated by short loops, and a cyclic permutation matrix coefficient a which is set to be '1' is needed in the specific operation process _ij The specific conditions to be met are that：

(1) Each time will a _ij When setting '1', the cyclic permutation matrix coefficient a with the largest number of short rings should be selected _ij ；

(2) At a _ij Under the condition of the same short ring, the cyclic permutation matrix coefficient a with the periphery containing "-1" least is arranged according to the sequence of the preceding column and the following column _ij Set to "-1";

(3) New set "-1" cyclic permutation matrix coefficient a _ij The number of the cyclic permutation coefficients which are connected with other cyclic permutation coefficients of (1) is not more than 3; unless the cyclic permutation matrix coefficient a _ij After setting '1', the number L of short rings can be eliminated ₁ And the cyclic permutation matrix coefficient a is except for _ij The maximum number of short loops L which can be eliminated by other cyclic permutation matrix coefficients ₂ Satisfy L therebetween ₁ -L ₂ 2 or more; and the eliminated rings are not counted again in the subsequent ring elimination process.

Preferably, the JACO-QC-LDPC code is used for obtaining the characteristic of large girth, and the specific steps are as follows:

step one, initializing, wherein x=1;

step two, H is carried out _v The most cyclic permutation matrix coefficient containing short rings in the x-th row of the matrix is set to be-1;

step three, when x is more than or equal to size (H) _b Step 1), when the step I is not satisfied, the step II is performed with resetting;

step four, H is carried out according to the judgment principle _b The cyclic permutation matrix coefficient with the largest number of short rings is set to be-1;

and fifthly, outputting a large girth JACO-QC-LDPC code.

Preferably, the constraint conditions in the process of performing the JACO-QC-LDPC code to obtain the large girth characteristic are:

(1) One cyclic permutation matrix coefficient a at a time _ij When the value is set to be '1', the corresponding eliminated short ring is not counted in any later traversal;

(2) Each a _ij At most 8 other cyclic permutation matrix coefficients are arranged on the periphery, and when the ring removing number sequences are the same, a is selected _ij Selecting peripheral no value of "-1" as much as possible when setting-1"a _ij ；

(3) New set of a of "-1 _ij The number of the connection between the front, the back, the left and the right is as small as possible and is not more than 3 unless the a _ij The number L of short rings can be eliminated after being set to be' 1 ₁ And in addition to the a _ij The maximum number of short loops L which can be eliminated by other cyclic permutation matrix coefficients ₂ Satisfy L therebetween ₁ -L ₂ ≥2；

(4) Search a _ij Traversing in the order of "antecedent-postamble" provided there are a plurality of a's of identical condition _ij When the ' 1 ' needs to be set, the sequence is prioritized, and once all a's are _ij If all the conditions cannot be satisfied, a with the least number of adjacent connections between the front, the back, the left and the right is selected _ij Set "-1".

The practical operation limitation is adopted, so that uneven distribution of the code codes is avoided, uneven weight of the code groups such as front heavy and rear light, front light and rear heavy, head-tail heavy and middle light and head-tail light and heavy are avoided, and the influence of uneven distribution of the code codes on the coding and decoding performance is avoided. Thus, the base matrix H is modified by mask aiming at JACO-QC-LDPC code _b And performing processing and operation of a ring elimination algorithm to generate the JACO-QC-LDPC code with short ring eliminated by the algorithm and 4 rings.

In simulation test comparison of coding and decoding performances under different code rates R, the method and the device respectively perform performance comparison on JACO-QC-LDPC codes with the code rates of R=1/2 and R=2/3 before and after modification and elimination of short rings. Time base matrix H when code rate is R=1/2 _b When the code rate is r=2/3, the base matrix is set to be 5×15, the spreading factor p=25 is set, 30 iterations of BP decoding algorithm are adopted, BPSK modulation is utilized, and simulation is carried out through a channel containing gaussian white noise.

Compared with the Fibonacci array, the differences among the elements of the Jacobsthal array do not show regular changes of the equal difference, the Jacobsthal array combination background is deep, and the Jacobsthal array combination background is particularly widely applied to the replacement mode problem and the grignard problem, and has quite deep combination background and application practical significance. Therefore, the base matrix structure adopted in the application is a double diagonal structure, and the base matrix of the QC-LDPC code is constructed by using Jacobsthal number columns. The base matrix adopts a double diagonal structure, the expression is identical to the expression (1-15), and the expression is as follows:

H _b ＝[H _v |H _c ] (1-15)

wherein H is _c Adopts the quasi-dual diagonal structure [30-31 ] in IEEE 802.16e]，H _c The expression of (2) is identical to the expressions (1-23) and (1-16), and the expressions (1-23) are:

for an (n, k) linear block code, the code rate R is expressed as:

the formula (1-16) is: h _b ＝[H _v |H _c ] _m×n ；

A fast iterative coding algorithm using a base matrix H _b The quasi-dual diagonal structure of (2) is directly and iteratively encoded by a check matrix H obtained after matrix expansion, and the iterative algorithm has lower encoding complexity, wherein, the base matrix H _b The structure of (2) satisfies the formulas (1-16), wherein H _c Adopts a quasi-dual diagonal structure in IEEE 802.16e, H _c The structure of (C) is shown as the formula (1-24), wherein h is satisfied _c (1)＝h _c (m _b ) The value range of r satisfies that r is more than or equal to 2 and less than or equal to m _b -1，H _v The structure of (C) is shown as the formula (1-25):

wherein h is satisfied with _c (1)＝h _c (m _b ) The value range of r satisfies that r is more than or equal to 2 and less than or equal to m _b -1，

In the construction of the base matrix, H _c The sizes of (2) are 5×5, let h _c (1)＝h _c (m _b )＝7，h _c (r) =0, and left submatrix H _v All elements of the first row of the (c) are configured as "0", the first column, and the elements of the first row other than the elements of the first row are numbers of the even number bits of the Jacobsthal number column, such as: 0. 1, 5, 21, … …, left subarray H _v Each row of numbers is a continuous Jacobsthal number column, H _v Except for the first row elements, and sequentially starting to sort backwards, thereby completing the QC-LDPC code left subarray H based on Jacobsthal array _v And finally obtaining the base matrix H of the QC-LDPC code based on Jacobsthal array _b . Wherein H is _v The expression of (2) is shown in the formula (1-17). In the article, when the code rate is 1/2, H _v The size is 5 multiplied by 5; when the code rate is 2/3, H _v The size is 5×10.

In order to deeply explore the performance quality degree and convenience performance comparison of the QC-LDPC code based on the Jacobsthal array structure, the method is called as a JACO-QC-LDPC code, and refers to the F-QC-LDPC code constructed by the Fibonacci array structure so as to explore the coding performance distinction between the JACO-QC-LDPC code and the F-QC-LDPC code, and the F-QC-LDPC code and the JACO-QC-LDPC code are subjected to performance comparison under the condition of no modification so as to explore the performance difference of the two coding modes under the condition of different signal to noise ratios. Setting a base matrix H _b The size is 5 multiplied by 10, the expansion factor p is 25, the code rate R is 1/2, the adopted decoding method is BP iterative decoding algorithm, the iteration number is set to 30, and the BPSK modulation mode is adopted to simulate through a channel containing Gaussian white noise.

The invention introduces a quick coding method and a plurality of decoding algorithms based on a quasi-dual diagonal structure design in IEEE 802.16e, and introduces the basic principle of a BP iterative decoding algorithm in detail through a Gallager soft decision decoding algorithm, utilizes the dual diagonal structure to construct QC-LDPC codes based on Jacobsthal sequences and Jacobsthal-Lucas sequences, and respectively performs performance comparison on the JACO-QC-LDPC codes before and after modification by a mask modification technology and F-QC-LDPC codes, thereby exploring the performance influence difference of the QC-LDPC codes based on the Jacobsthal sequences. Finally, according to the short ring counting and detection elimination algorithm researched by the invention, aiming at the quasi-cyclic characteristics of the QC-LDPC code, the QC-LDPC based on the Jacobsthal array and the Jacobsthal-Lucas array and obtained after the short ring and 4-ring elimination algorithm is designed, the performance improvement is realized, and finally, the expanded tree structure of the Tanner graph based on the QC-LDPC code base matrix is designed by means of the short ring counting and search concept, so that the 4-ring and 6-ring counting and elimination are carried out, the large girth JACO-QC-LDPC code based on the Jacobsthal array is constructed on the basis, and the second improvement of the QC-LDPC code coding performance is realized again.

The innovation points of the invention are as follows:

(1) The QC-LDPC code based on the Jacobsthal array is creatively designed, compared with the F-QC-LDPC code, the quick coding and the improvement of the coding performance are realized, and the method has better feasibility.

(2) A QC-LDPC code based on a Jacobsthal-Lucas sequence is innovatively designed, and qualitative explanation is made on the performance stability of the QC-LDPC code.

(3) The short loop elimination is directly carried out on the JACO-QC-LDPC code and the JACO-LUCAS-QC-LDPC code after mask modification, so that the coding performance of the flexible code group with any code length is improved once.

(4) An algorithm for counting and eliminating 4-ring and 6-ring is innovatively designed, and a JACO-QC-LDPC code with large girth is successfully constructed.

The invention researches and discovers the structure of the large girth JACO-QC-LDPC code based on the Jacobsthal array, establishes a novel base matrix architecture, ring elimination modification and algorithm method, and finally constructs a code group with multiple functions of code number, code length, flexible code rate selection, quick coding, low error rate and the like, which is a good code and is a array coding scheme worthy of research.

The specific embodiment is as follows:

the construction method of the QC-LDPC code based on the Jacobsthal array comprises the following steps:

constructing a QC-LDPC code of a dual diagonal basis matrix structure by using Jacobsthal arrays; the design can generate a short-loop elimination method with large girth, and the large girth QC-LDPC code based on Jacobsthal number series is constructed by modifying the cyclic permutation matrix coefficient.

Jacobsthal array { J _n Defined as:

J _n ＝J _n-1 +2J _n-2 (1-1)

in the formula (1-1) { J _n First two items J ₀ And J ₁ The formula II is as follows:

the recurrence formula for the Fibonacci array F (n), also known as the golden section array, is:

F(0)＝0 F(1)＝1 (1-3)

F(n)＝F(n-1)+F(n-2) (n≥2,n∈N ^* ) (1-4)

formulae (1-4) and (1-5) can be successively deduced { J } _n The } antecedents are respectively: 0. 1, 3, 5, 11, … …, however, if one needs to be found directly, the first few specific values need to be known in advance, which is useful for determining { J ] _n The term value of the specified term of the sequence is quite inconvenient, so that a common generating function phi (t) of the sequence needs to be determined to obtain a recurrence relation, wherein the expression of the recurrence generating function phi (t) is shown as the following formula (1-6):

when the formula (1-7) is satisfied:

then for any real number c, it satisfies the equation (1-8) as:

let Jacobsthal number column { J _n The occurrence function Φ (t) satisfying the equation (1-11) is:

then the formula (1-12) is:

the final formulae (1-13) and (1-14) are:

the base matrix structure is a double diagonal structure, a base matrix of the QC-LDPC code is constructed by using Jacobsthal number columns, and the base matrix adopts the double diagonal structure and has the expression:

H _b ＝[H _v |H _c ] (1-15)

wherein H is _c Adopts a quasi-dual diagonal structure in IEEE 802.16e, H _c The expression of (2) is the formula (1-16), wherein h is satisfied _c (1)＝h _c (m _b ) The value range of r satisfies that r is more than or equal to 2 and less than or equal to m _b -1；

In the construction of the base matrix, H _c The sizes of (2) are 5×5, let h _c (1)＝h _c (m _b )＝7，h _c (r) =0, and left submatrix H _v All elements of the first row of (a) are configured as "0", the first column is the head of the rest of the rows except the elements of the first row, the elements of the first column are the digits of the even number of the Jacobsthal number column, such as: 0. 1, 5, 21, … …, left subarray H _v Each row of numbers is a continuous Jacobsthal number column, H _v Except for the first row elements, and sequentially starting to sort backwards, thereby completing the QC-LDPC code left subarray H based on Jacobsthal array _v And finally obtaining the base matrix H of the QC-LDPC code based on Jacobsthal array _b Wherein H is _v The expression of (2) is shown as the expression (1-17), when the code rate is 1/2, H _v The size is 5 multiplied by 5; when the code rate is 2/3, H _v The size is 5 multiplied by 10;

the QC-LDPC code based on the Jacobsthal array structure is called as a JACO-QC-LDPC code, and the JACO-QC-LDPC code eliminates a short ring, and the specific method is as follows: traversing according to a 4-ring method, and designing a unit cyclic permutation matrix

Knowing the cyclic permutation matrix coefficient a _i,j Cyclic permutation matrix sizeFor p×p, p is the spreading factor, then the non-zero element, element "1", in the cyclic permutation matrix is located at a position (0, a _i,j )、(1,a _i,j +1)、(2,a _i,j +2)、…、(p-a _i,j ,p)、(p-a _i,j +1,1)、…、(p-1,a _i,j -1) wherein the cyclic permutation matrix +.>

As a base matrix H _b Is expressed by the following formula

And satisfies i=1, 2, …, m; j=1, 2, …, n, as can be seen from the above, when a _i,j Not equal to-1, i.e. the cyclic permutation matrix

For a non-zero matrix, <' > a matrix of zero>

The non-zero element, element "1", is present in the position (x, mod [ (x+a) _i,j ),p]) Where x=0, 1, …, p-1 is satisfied;

setting information forming the existence of the 4-ring and circulating replacement matrix where check nodes are located

Corresponding coefficients of (a) are respectively

Assuming that the positions of the information nodes and the check nodes constituting the 4-ring are (m ₁ ,n ₁ )、(m ₂ ,n ₂ )、(m ₃ ,n ₃ )、(m ₄ ,n ₄ ) From the element positions of the respective information nodes and check nodes, it is possible and prescribed according to the formulas (1-18):

according to

The non-zero element, element "1", is present in the position (x, mod [ (x+a) _i,j ),p]) The method can obtain:

the formulae (1-19) and (1-20) are jointly obtainable:

finally, the formula (1-22) is simplified to obtain:

due to the base matrix H _b Smaller, so that the base matrix H can be directly modified by the masking technology according to the specific shape of the 4-ring _b Traversing by adopting a traversing search 4-ring method, and obtaining 4 cyclic permutation matrix coefficients a by single traversing _i,j Substituting the four cyclic permutation matrices into the expression (1-22) for verification, and if the expression (1-22) is satisfied, recognizing the four cyclic permutation matrices

Each information node is provided with 1 information node to form a 4-ring, the searched 4-rings are stored, and then the key cyclic permutation matrix coefficient a is carried out _ij Setting to-1 to complete the +.>

The 4 rings existing in the inner part are eliminated, so that the purpose of eliminating the rings is achieved.

Eliminating short rings of JACO-QC-LDPC codes, in particularThe cyclic permutation matrix coefficient a which needs to be set to '1' in the operation process _ij The specific conditions to be met are:

(1) Each time will a _ij When setting '1', the cyclic permutation matrix coefficient a with the largest number of short rings should be selected _ij ；

(2) At a _ij Under the condition of the same short ring, the cyclic permutation matrix coefficient a with the periphery containing "-1" least is arranged according to the sequence of the preceding column and the following column _ij Set to "-1";

(3) New set "-1" cyclic permutation matrix coefficient a _ij The number of the cyclic permutation coefficients which are connected with other cyclic permutation coefficients of (1) is not more than 3; unless the cyclic permutation matrix coefficient a _ij After setting '1', the number L of short rings can be eliminated ₁ And the cyclic permutation matrix coefficient a is except for _ij The maximum number of short loops L which can be eliminated by other cyclic permutation matrix coefficients ₂ Satisfy L therebetween ₁ -L ₂ 2 or more; and the eliminated rings are not counted again in the subsequent ring elimination process.

The JACO-QC-LDPC code is used for obtaining the characteristic of large girth, and the specific steps are as follows:

step one, initializing, wherein x=1;

step two, H is carried out _v The most cyclic permutation matrix coefficient containing short rings in the x-th row of the matrix is set to be-1;

step three, when x is more than or equal to size (H) _b Step 1), when the step I is not satisfied, the step II is performed with resetting;

step four, H is carried out according to the judgment principle _b The cyclic permutation matrix coefficient with the largest number of short rings is set to be-1;

and fifthly, outputting a large girth JACO-QC-LDPC code.

The limiting conditions in the process of performing JACO-QC-LDPC code to obtain large girth characteristics are as follows:

(1) One cyclic permutation matrix coefficient a at a time _ij When the value is set to be '1', the corresponding eliminated short ring is not counted in any later traversal;

(2) Each a _ij At most 8 other cyclic permutation matrix coefficients are arranged on the periphery, and the number of the cyclic permutation matrix coefficients is removedSelecting a certain a when the columns are the same _ij When setting-1, selecting a with peripheral no value of minus 1 as much as possible _ij ；

(3) New set of a of "-1 _ij The number of the connection between the front, the back, the left and the right is as small as possible and is not more than 3 unless the a _ij The number L of short rings can be eliminated after being set to be' 1 ₁ And in addition to the a _ij The maximum number of short loops L which can be eliminated by other cyclic permutation matrix coefficients ₂ Satisfy L therebetween ₁ -L ₂ ≥2；

(4) Search a _ij Traversing in the order of "antecedent-postamble" provided there are a plurality of a's of identical condition _ij When the ' 1 ' needs to be set, the sequence is prioritized, and once all a's are _ij If all the conditions cannot be satisfied, a with the least number of adjacent connections between the front, the back, the left and the right is selected _ij Set "-1".

The practical operation limitation is adopted, so that uneven distribution of the code codes is avoided, uneven weight of the code groups such as front heavy and rear light, front light and rear heavy, head-tail heavy and middle light and head-tail light and heavy are avoided, and the influence of uneven distribution of the code codes on the coding and decoding performance is avoided. Thus, the base matrix H is modified by mask aiming at JACO-QC-LDPC code _b And performing processing and operation of a ring elimination algorithm to generate the JACO-QC-LDPC code with short ring eliminated by the algorithm and 4 rings.

In simulation test comparison of coding and decoding performances under different code rates R, the method and the device respectively perform performance comparison on JACO-QC-LDPC codes with the code rates of R=1/2 and R=2/3 before and after modification and elimination of short rings. Time base matrix H when code rate is R=1/2 _b When the code rate is r=2/3, the base matrix is set to be 5×15, the spreading factor p=25 is set, 30 iterations of BP decoding algorithm are adopted, BPSK modulation is utilized, and simulation is carried out through a channel containing gaussian white noise. Tables 1-1 and 1-2 show BER statistics of small signal to noise ratios before and after elimination of the short loop for JACO-QC-LDPC codes at different code rates, respectively.

Table 1-1 partial small snr BER statistical comparison at code rate r=1/2

Table 1-2 partial small snr BER statistical comparison at code rate r=2/3

Compared with the Fibonacci array, the differences among the elements of the Jacobsthal array do not show regular changes of the equal difference, the Jacobsthal array combination background is deep, and the Jacobsthal array combination background is particularly widely applied to the replacement mode problem and the grignard problem, and has quite deep combination background and application practical significance. Therefore, the base matrix structure adopted in the application is a double diagonal structure, and the base matrix of the QC-LDPC code is constructed by using Jacobsthal number columns. The base matrix adopts a double diagonal structure, the expression is identical to the expression (1-15), and the expression is as follows:

H _b ＝[H _v |H _c ] (1-15)

wherein H is _c Adopts the quasi-dual diagonal structure [30-31 ] in IEEE 802.16e]，H _c The expression of (2) is identical to the expressions (1-23) and (1-16), and the expressions (1-23) are:

for an (n, k) linear block code, the code rate R is expressed as:

the formula (1-16) is: h _b ＝[H _v |H _c ] _m×n ；

A fast iterative coding algorithm using a base matrix H _b Is directly and iteratively encoded by a check matrix H obtained after matrix expansion, and has lower encoding complexity, whereinBase matrix H _b The structure of (2) satisfies the formulas (1-16), wherein H _c Adopts a quasi-dual diagonal structure in IEEE 802.16e, H _c The structure of (C) is shown as the formula (1-24), wherein h is satisfied _c (1)＝h _c (m _b ) The value range of r satisfies that r is more than or equal to 2 and less than or equal to m _b -1，H _v The structure of (C) is shown as the formula (1-25):

wherein h is satisfied with _c (1)＝h _c (m _b ) The value range of r satisfies that r is more than or equal to 2 and less than or equal to m _b -1，

In the construction of the base matrix, H _c The sizes of (2) are 5×5, let h _c (1)＝h _c (m _b )＝7，h _c (r) =0, and left submatrix H _v All elements of the first row of the (c) are configured as "0", the first column, and the elements of the first row other than the elements of the first row are numbers of the even number bits of the Jacobsthal number column, such as: 0. 1, 5, 21, … …, left subarray H _v Each row of numbers is a continuous Jacobsthal number column, H _v Except for the first row elements, and sequentially starting to sort backwards, thereby completing the QC-LDPC code left subarray H based on Jacobsthal array _v And finally obtaining the base matrix H of the QC-LDPC code based on Jacobsthal array _b . Wherein H is _v The expression of (2) is shown in the formula (1-17). In the article, when the code rate is 1/2, H _v The size is 5 multiplied by 5; when the code rate is 2/3, H _v The size is 5×10.

In order to deeply explore the performance quality degree and convenience performance comparison of the QC-LDPC code based on the Jacobsthal array structure, the method is called as a JACO-QC-LDPC code, and refers to the F-QC-LDPC code constructed by the Fibonacci array structure so as to explore the coding performance distinction between the JACO-QC-LDPC code and the F-QC-LDPC code, and the F-QC-LDPC code and the JACO-QC-LDPC code are subjected to performance comparison under the condition of no modification so as to explore the performance difference of the two coding modes under the condition of different signal to noise ratios. Setting a base matrix H _b The size is 5 multiplied by 10, the expansion factor p is 25, the code rate R is 1/2, the adopted decoding method is BP iterative decoding algorithm, the iteration number is set to 30, and the BPSK modulation mode is adopted to simulate through a channel containing Gaussian white noise. Tables 1-3 are BER statistics for unmodified F-QC-LDPC codes and JACO-QC-LDPC codes with small signal-to-noise ratios.

Tables 1-3 comparison of small signal to noise ratio partial BER of unmodified JACO-QC-LDPC code with F-QC-LDPC code

The invention introduces a quick coding method and a plurality of decoding algorithms based on a quasi-dual diagonal structure design in IEEE 802.16e, and introduces the basic principle of a BP iterative decoding algorithm in detail through a Gallager soft decision decoding algorithm, utilizes the dual diagonal structure to construct QC-LDPC codes based on Jacobsthal sequences and Jacobsthal-Lucas sequences, and respectively performs performance comparison on the JACO-QC-LDPC codes before and after modification by a mask modification technology and F-QC-LDPC codes, thereby exploring the performance influence difference of the QC-LDPC codes based on the Jacobsthal sequences. Finally, according to the short ring counting and detection elimination algorithm researched by the invention, aiming at the quasi-cyclic characteristics of the QC-LDPC code, the QC-LDPC based on the Jacobsthal array and the Jacobsthal-Lucas array and obtained after the short ring and 4-ring elimination algorithm is designed, the performance improvement is realized, and finally, the expanded tree structure of the Tanner graph based on the QC-LDPC code base matrix is designed by means of the short ring counting and search concept, so that the 4-ring and 6-ring counting and elimination are carried out, the large girth JACO-QC-LDPC code based on the Jacobsthal array is constructed on the basis, and the second improvement of the QC-LDPC code coding performance is realized again.

The innovation points of the invention are as follows:

(1) The QC-LDPC code based on the Jacobsthal array is creatively designed, compared with the F-QC-LDPC code, the quick coding and the improvement of the coding performance are realized, and the method has better feasibility.

(2) A QC-LDPC code based on a Jacobsthal-Lucas sequence is innovatively designed, and qualitative explanation is made on the performance stability of the QC-LDPC code.

(3) The short loop elimination is directly carried out on the JACO-QC-LDPC code and the JACO-LUCAS-QC-LDPC code after mask modification, so that the coding performance of the flexible code group with any code length is improved once.

(4) An algorithm for counting and eliminating 4-ring and 6-ring is innovatively designed, and a JACO-QC-LDPC code with large girth is successfully constructed.

The invention researches and discovers the structure of the large girth JACO-QC-LDPC code based on the Jacobsthal array, establishes a novel base matrix architecture, ring elimination modification and algorithm method, and finally constructs a code group with multiple functions of code number, code length, flexible code rate selection, quick coding, low error rate and the like, which is a good code and is a array coding scheme worthy of research.

Claims (2)

1. A construction method of QC-LDPC codes based on Jacobsthal arrays is characterized by comprising the following steps: the construction method of the QC-LDPC code based on the Jacobsthal array comprises the following steps:

constructing a QC-LDPC code of a dual diagonal basis matrix structure by using Jacobsthal arrays; designing a short-loop elimination method capable of generating large girth, and constructing a large girth QC-LDPC code based on Jacobsthal sequence by modifying a cyclic permutation matrix coefficient;

jacobsthal array { J _n Defined as:

J _n ＝J _n-1 +2J _n-2 (1-1)

in the formula (1-1) { J _n First two items J ₀ And J ₁ The formula II is as follows:

the recurrence formula for the Fibonacci array F (n), also known as the golden section array, is:

F(0)＝0 F(1)＝1 (1-3)

F(n)＝F(n-1)+F(n-2) (n≥2,n∈N ^* ) (1-4)

formulae (1-4) and (1-5) can be successively deduced { J } _n The } antecedents are respectively: 0. 1, 3, 5, 11, … …, however, if one needs to be found directly, the first few specific values need to be known in advance, which is useful for determining { J ] _n The term value of the specified term of the sequence is quite inconvenient, so that a common generating function phi (t) of the sequence needs to be determined to obtain a recurrence relation, wherein the expression of the recurrence common generating function phi (t) is shown as the following formula (1-6):

when the formula (1-7) is satisfied:

then for any real number c, it satisfies the equation (1-8) as:

let Jacobsthal number column { J _n The normal occurrence function Φ (t) satisfies the equation (1-11):

then the formula (1-12) is:

the final formulae (1-13) and (1-14) are:

the base matrix structure is a double diagonal structure, a base matrix of the QC-LDPC code is constructed by using Jacobsthal number columns, and the base matrix adopts the double diagonal structure and has the expression:

H _b ＝[H _v |H _c ] (1-15)

wherein H is _c Adopts a quasi-dual diagonal structure in IEEE 802.16e, H _c The expression of (2) is the formula (1-16), wherein h is satisfied _c (1)＝h _c (m _b ) The value range of r satisfies that r is more than or equal to 2 and less than or equal to m _b -1；

In the construction of the base matrix, H _c The sizes of (2) are 5×5, let h _c (1)＝h _c (m _b )＝7，h _c (r) =0, and left submatrix H _v All elements of the first row of (a) are configured as "0", the first column is the head of the rest of the rows except the elements of the first row, the elements of the first column are the digits of the even number of the Jacobsthal number column, such as: 0. 1, 5, 21, … …, left submatrix H _v Each row of numbers is a continuous Jacobsthal number column, H _v Except for the first row elements, and sequentially starting to sort backwards, thereby completing the QC-LDPC code left submatrix H based on Jacobsthal number columns _v And finally obtaining the base matrix H of the QC-LDPC code based on Jacobsthal array _b Wherein H is _v The expression of (2) is shown as the expression (1-17), when the code rate is 1/2, H _v The size is 5 multiplied by 5; when the code rate is 2/3, H _v The size is 5 multiplied by 10;

the QC-LDPC code based on the Jacobsthal array structure is called as a JACO-QC-LDPC code, and the JACO-QC-LDPC code eliminates a short ring, and the specific method is as follows: traversing according to a 4-ring method, and designing a unit cyclic permutation matrix

Knowing the cyclic permutation matrix coefficient a _i,j The cyclic permutation matrix has a size p×p, p is the spreading factor, and the non-zero element, i.e., element "1", in the cyclic permutation matrix is located at a position (0, a) _i,j )、(1,a _i,j +1)、(2,a _i,j +2)、…、(p-a _i,j ,p)、(p-a _i,j +1,1)、…、(p-1,a _i,j -1) wherein the unit cyclic permutation matrix +.>

As a base matrix H _b Is expressed by the following formula

And satisfies i=1, 2, …, m; j=1, 2, …, n, as can be seen from the above, when a _i,j Not equal to-1, i.e. unit cyclic permutation matrix

For a non-zero matrix, <' > a matrix of zero>

The non-zero element, element "1", is present in the position (x, mod [ (x+a) _i,j ),p]) Where x=0, 1, …, p-1 is satisfied;

setting information forming the existence of the 4-ring and unit cyclic permutation matrix where check nodes are located

Corresponding coefficients of (a) are respectively

Assume that the positions in each unit cyclic permutation matrix where the information nodes and the check nodes constituting the 4-ring are located are (m ₁ ,n ₁ )、(m ₂ ,n ₂ )、(m ₃ ,n ₃ )、(m ₄ ,n ₄ ) From the element positions of the respective information nodes and check nodes, it is possible and prescribed according to the formulas (1-18):

according to

The non-zero element, element "1", is present in the position (x, mod [ (x+a) _i,j ),p]) The method can obtain:

the formulae (1-19) and (1-20) are jointly obtainable:

finally, the formula (1-21) is simplified to obtain:

due to the base matrix H _b Smaller, so that the base matrix H can be directly modified by the masking technology according to the specific shape of the 4-ring _b Traversing by adopting a traversing search 4-ring method, and obtaining 4 cyclic permutation matrix coefficients a by single traversing _i,j Substituting the matrix into the formula (1-22) for verification, and if the formula (1-22) is satisfied, recognizing four unit cycle permutation matrices

Each information node is provided with 1 information node to form a 4-ring, the searched 4-rings are stored, and then the key cyclic permutation matrix coefficient a is carried out _ij Setting to-1 to complete the +.>

Eliminating the 4 rings existing in the inner part, so that the purpose of eliminating the rings is achieved;

eliminating short rings of JACO-QC-LDPC codes, and setting the cyclic permutation matrix coefficient a of "-1" in the specific operation process _ij The specific conditions to be met are:

(1) Each time will a _ij When setting '1', the cyclic permutation matrix coefficient a with the largest number of short rings should be selected _ij ；

(2) At a _ij Under the condition of the same short ring, the cyclic permutation matrix coefficient a with the periphery containing "-1" least is arranged according to the sequence of the preceding column and the following column _ij Set to "-1";

(3) New set "-1" cyclic permutation matrix coefficient a _ij And to thisHe is "-1" and the cyclic permutation coefficients must not be connected by more than 3; unless the cyclic permutation matrix coefficient a _ij After setting '1', the number L of short rings can be eliminated ₁ And the cyclic permutation matrix coefficient a is except for _ij The maximum number of short loops L which can be eliminated by other cyclic permutation matrix coefficients ₂ Satisfy L therebetween ₁ -L ₂ 2 or more; and the eliminated rings are not counted again in the subsequent ring elimination process;

the JACO-QC-LDPC code is used for obtaining the characteristic of large girth, and the specific steps are as follows:

step one, initializing, wherein x=1;

step two, H is carried out _v The most cyclic permutation matrix coefficient containing short rings in the x-th row of the matrix is set to be-1;

step three, when x is more than or equal to size (H) _b Step 1), when the step I is not satisfied, the step II is performed with resetting;

step four, H is carried out according to the judgment principle _b The cyclic permutation matrix coefficient with the largest number of short rings is set to be-1;

and fifthly, outputting a large girth JACO-QC-LDPC code.

2. The construction method of QC-LDPC codes based on Jacobsthal arrays as claimed in claim 1, wherein: the limiting conditions in the process of performing JACO-QC-LDPC code to obtain large girth characteristics are as follows:

(1) One cyclic permutation matrix coefficient a at a time _ij When the value is set to be '1', the corresponding eliminated short ring is not counted in any later traversal;

(2) Each a _ij At most 8 other cyclic permutation matrix coefficients are arranged on the periphery, and when the ring removing number sequences are the same, a is selected _ij When setting-1, selecting a with peripheral no value of minus 1 as much as possible _ij ；

(3) New set of a of "-1 _ij The number of the connection between the front, the back, the left and the right is as small as possible and is not more than 3 unless the a _ij The number L of short rings can be eliminated after being set to be' 1 ₁ Infinity, and besides a _ij The maximum number of short loops L which can be eliminated by other cyclic permutation matrix coefficients ₂ Satisfy L therebetween ₁ -L ₂ ≥2；

(4) Search a _ij Traversing in the order of "antecedent-postamble" provided there are a plurality of a's of identical condition _ij When the ' 1 ' needs to be set, the sequence is prioritized, and once all a's are _ij If all the conditions cannot be satisfied, a with the least number of adjacent connections between the front, the back, the left and the right is selected _ij Set "-1".