CN106998208A - A kind of code construction method of variable length Polar codes - Google Patents

Abstract

The invention provides a kind of code construction method of variable length Polar codes, belong to the communications field.The present invention realizes the construction of long code generator matrix by extending the generator matrix of short code, is primarily based on integer binary representation, code length is expressed as to the summation form of 2 power；Then will constitute summation form each single item value as the submatrix for constituting newly-generated matrix size；Finally, new generator matrix is constructed by combo sum combinatorial operations.Compared with the building method of the long Polar codes of conventional variable, the present invention need not punch deletion action, reduce encoder complexity, reduce time delay, at decoding end for known to the likelihood information of known bits, the problem of traditional conventional method estimates known bits likelihood information using priori is avoided, so as to reduce the bit error rate of decoding, systematic function is improved.

Description

A kind of code construction method of variable length Polar codes

Technical field

The invention belongs to communication channel coding field, a kind of code construction method of variable length Polar codes is specifically referred to.

Background technology

Polar Codes, i.e. polarization code are 2009 by E.A kind of new channel coding proposed.Polarization code It is designed based on channel-polarization (Channel Polarization), is that the first can be proved by strict mathematical method Reach the constructivity encoding scheme of channel capacity.However, polarization code is constructed by Kronecker powers, this make is limited The code length of Polar codes, is not easy to the use of Polar codes in systems in practice.It is 2 that original method, which can only construct code length,ⁿ(n=1, 2 ...) Polar codes.Although the Polar codes of other code lengths other polarized nucleus can be constructed by BCH code core etc., this The Polar codes code length of method construct is still limited in the power of core length, and the decoding knot of the Polar codes of this method construct Structure is complex.

Well known in the art, in practical communication system, the length of raw information is often uncertain, and this requires to compile Code can be adjusted according to the length of raw information.Obtained according to the length of channel condition and information with a series of differences The coding structure of code length code check.The code construction of current existing variable length polarization code is used to power that original length is 2 Secondary code word bits delete partial codeword bit to realize the code construction of non-2 power.This method is when receiving terminal is decoded pair 0,1 equiprobability is set in the likelihood information for the code word bits deleted, the decoding of common polarization code is carried out to it again afterwards.The party Although method realizes the Polar codes construction of variable code length, but its decoding error probability is also enhanced, heavy losses communication system Performance.Therefore, a kind of code construction method of the variable Polar codes of the more preferable code length of performance is a kind of demand.

The content of the invention

Based on the demand, the present invention proposes a kind of code construction method of variable length Polar codes, original length is defined to 2 power Polar code generator matrixes are combined to realize the construction of the generator matrix of the Polar codes of any code length.

A kind of code construction method for variable length Polar codes that the present invention is provided, including step are as follows：

Step 1：To code length N binary expansions, corresponding binary bit sequence s is obtained；Wherein N is positive integer；

Step 2：The position of " 1 " in the binary bit sequence s obtained according to step 1, it is determined that needing the generation square chosen The size of battle array.

If bit value is q for the quantity of " 1 " in sequence s, index number h in wherein sequence s_dBit value represent for 1, d D-th of 1, d=1,2 ... the q occurred in sequence s from a high position to low level.That then chooses q polar code is originally generated matrix, divides It is not that size is l₁,l₂,...,l_qSquare formation,h_dFor positive integer, sequence s is from low level to a high position from 1 Proceed by index number.

Step 3：The submatrix of above-mentioned selection is combined by Combo-Sum computings, so as to obtain the generation square that code length is N Battle array.

If obtaining q generator matrix according to step 2Then code length represents for N generator matrix C For：

Wherein,Represent Combo-Sum computings.

Described Combo-Sum computings are：If the matrix for participating in computing isWithObtaining combinatorial matrix is：

Wherein, L₁、L₂Respectively matrix A, B size, matrix B ' size are L₂×L₁, the preceding L in B '₂Row and matrix B phase Together, rear L₁-L₂It is classified as full 0 row.

Advantages of the present invention is with good effect：The code construction method of the present invention is by Combo-Sum computings come real Existing, the Polar codes that this method is constructed can be such that decoding error probability effectively reduces, and lift communication system performance.The present invention passes through Extend the generator matrix of short code to realize the construction of long code generator matrix, punch operation need not be deleted compared with conventional method, Encoder complexity is reduced, time delay is reduced.The present invention is decoding likelihood information of the end for known bits, it is known that avoiding traditional biography The problem of system method estimates known bits likelihood information using priori, so as to reduce the bit error rate of decoding, lifting system Performance.

Brief description of the drawings

Fig. 1 is the schematic flow sheet of code construction method proposed by the invention；

Fig. 2 for the present invention construction generator matrix used in the process of to three specific examples for being originally generated matrix；

Fig. 3 is the schematic diagram for generally corresponding to relation of newly-generated matrix and submatrix of the invention.

Embodiment

Below in conjunction with drawings and examples, the present invention is described in further detail.

As shown in figure 1, the code construction method of variable length Polar codes of the present invention, real by below step 1~3 The construction of existing Polar code generator matrixes.

Step 1：Binary bit sequence is obtained for code length N by binary expansion to represent.N is positive integer.

For code length N, m bit-binary bit sequence s={ s are obtained with binary expansion_m,s_m-1,...,s₂,s₁, if h It is binary bit sequence from low level to high-order index number, h=1,2 ... m, s_hRepresent that index position is h's in sequence s Bit value, s_h∈ { 0,1 }, m represent the index position of binary sequence highest order 1, s_m=1.

Here code length N can take any positive integer, it is not necessary to as the code length of original polar codes can only be defined to 2 power Form.

For example as code length N=13, corresponding binary sequence s={ 1,1,0,1 }.

Step 2：Combinatorial operation generation square to be chosen is determined according to the position of " 1 " in obtained binary expansion sequence Battle array.

The position that bit value in binary expansion sequence is " 1 " is found out, if the quantity of " 1 " is q.Two according to obtained by step 1 System deploys sequence, then N can be expressed as：

Formula (1) represents the position according in binary bit sequence 1, and N is expressed as to 2 power form, wherein, sequence Index number h in s_dBit value represent in sequence s to occur from a high position to low level for 1, d d-th 1.

That then chooses q polar code is originally generated matrix, wherein the size of d-th of generator matrix is l_dIt is natural number and the power form for 2.

Choose q l corresponding to the disassembly in the binary system with N_d×l_dPolar codes be originally generated matrix as combination transport Submatrix to be chosen is calculated, the submatrix of selection is expressed asWhereinRepresent l_d×l_dSquare formation, MatrixMatrix is originally generated for Polar codes, is produced by Kronecker product, i.e.,Wherein matrixParameter n=log₂l_d,Represent in Crow can product calculation, product calculation and square can be originally generated in the Crow Battle array is recognized by field.

For example：As code length N=6,6=2²+2¹=l₁+l₂, l₁=4, l₂=2, then choosing original Polar codes code length is 4 generator matrix G_4×4It is 2 generator matrix G with code length_2×2As submatrix of the code length for 6 generator matrix is constituted, square is generated Battle array G_4×4And G_2×2Occurrence represents as shown in Figure 2.

As code length N=8,8=2³=l₁, it can be seen that original length is the Polar code generator matrix structures of 2 power Make a special case of simply the inventive method, generator matrix G_8×8As shown in Figure 2.

Step 3：The generator matrix of above-mentioned selection is combined by Combo-Sum computings, so as to construct the generation that code length is N Matrix.

Matrix C ombo-Sum operation definitions are proposed first.Combo-Sum is the combinatorial operation of two matrixes, matrix A and square Battle array B Combo-Sum sums C can be expressed asMatrixWithIt is square formation, L₁、L₂Respectively square Battle array A, B size.In order to which the size for obtaining arbitrary size C, A and B need not be identical, L is taken here₁≥L₂.Matrix B in formula ' remove Identical with matrix B beyond full 0 row, B ' sizes are L₂×L₁, wherein, B ' preceding L₂Arrange, rear L identical with the row in matrix B₁- L₂It is classified as full 0 row.

The code word obtained by Matrix C as generator matrix can be expressed as x=uC, and x and u are further then decomposed into two Part, is expressed as x=(x₁,x₂) and u=(u₁,u₂), x represents the coding codeword vector obtained by generator matrix, x₁Represent Preceding L₁Individual code word,x₂L after expression₂Individual code word,u₁Presentation code is inputted The preceding L of sequence₁Individual bit,u₂The rear L of presentation code list entries₂Individual bit,According to the structure of Matrix C, x₁By u₁And u₂Obtained jointly by matrix A and B 'x₂By u₂X is obtained by matrix B₂=u₂B, so generation code word x can be expressed as

Combo-Sum operation definitions：Give two square formationsWherein i representing matrixs Line index, the column index of j representing matrixs, L₁,L₂The size of expression matrix, and L₁≥L₂.Define two matrixes and carry out Combo- Sum computings are combined Combo-Sum combinatorial operations are represented, wherein：

The c that formula (2) is represented_ijWith a_ijAnd b_ijCorresponding specific bit position it is as shown in Figure 3.Wherein, Γ be 1, 2,...,L₁A subset, referred to as difference collection, set element number is L₂, Γ be matrix B ' non-zero column set, necessarily Take B ' preceding L₂Row, but value order be not necessarily by 1,2 ... order arrangement.If index marker is j in set Γ₀=1, 2,...,L₂, index j is the jth in set Γ in Matrix C₀Individual element, j is determined according to j from Γ₀, obtain c_ijCorresponding square Element in battle array B

WhenCombo-Sum and C can be expressed as

The corresponding bit position that formula (3) is represented is as shown in Figure 3.

For example

Combinatorial operationHave defined below：

1. the A of computing is participated in, B matrixes cannot exchange sequence.

2. the size of first matrix can not be less than second matrix, i.e. l₁≥l₂。

3. Combo-Sum operation results are not unique.Different difference collection Γ and order of operation can obtain different Combo- Sum and.

Construction for example for code length for 5 generator matrix can have following two kinds of structures mode：

Make one：

Make two：

Long code can resolve into two or more short codes, because Combo-Sum computings are not unique, and step 3 is it is determined that it is joined During with the submatrixs of Combo-Sum computings, the disassembly in the binary system order that is obtained in strict accordance with code length N by step 2 and shape is represented Formula.

It can only be such as expressed as N=5 generator matrixAnd can not be expressed asWithCombining form or other representations.

So, the calculating process of step 3 is that q obtained according to step 2 are originally generated matrix Combo-Sum combinatorial operations are carried out to it, are provided during combinatorial operation, every time two maximum generator matrixes of selection code length Participate in Combo-Sum combinatorial operations, i.e., it is N's to be originally generated the code length that matrix obtained by Combo-Sum computings according to q Generator matrix

For example, code length N=7 generator matrix combinatorial operation process is：Obtain participating in the generation of combinatorial operation according to step 2 Matrix is G_4×4,G_2×2,G_1×1, according to operational criterion：Two maximum generator matrixes of selection participate in computing every time, thenMatrix combination is expressed as：

The combinatorial matrix that each Combo-Sum computings are obtained is used as a new submatrix, its son of replacement combination producing Matrix, continues to participate in next round Combo-Sum computings, until being only left a submatrix, obtains the generation that final code length is N Matrix C.

Embodiment 1：The generator matrix construction process of code length N=6 Polar codes.

Step 1：Code length N=6 binary bit sequence is expressed as s={ 1,1,0 }.

Step 2：The binary bit sequence obtained according to step 1, chooses the combinatorial matrix for constituting code length N=6：

According to 6=2²+2¹=l₁+l₂, so it is G to obtain constituting the submatrix that code length is 6_4×4,G_2×2。

Step 3：Pass through the sub- square of the generator matrix of the composition code length N=6 obtained by Combo-Sum computings combination step 2 Battle array.Calculating process is as follows：

Embodiment 2：The generator matrix construction process of code length N=11 Polar codes.

Step 1：Code length N=11 binary bit sequence is expressed as s={ 1,0,1,1 }.

Step 2：The binary bit sequence obtained according to step 1, chooses the combinatorial matrix for constituting code length N=11：

According to 11=2³+2¹+2⁰=l₁+l₂+l₃, the size of submatrix is l₁=8, l₂=2, l₃=1.So being constituted The submatrix that code length is 11 is G_8×8,G_2×2,G_1×1。

Step 3：Pass through the sub- square of the generator matrix of the composition code length N=11 obtained by Combo-Sum computings combination step 2 Battle array.Calculating process is as follows：

Claims (3)

1. a kind of variable Polar code code construction methods of code length, it is characterised in that comprise the following steps：

Step 1：To code length N binary expansions, corresponding binary bit sequence s is obtained；Wherein N is positive integer；

Step 2：According to the position of " 1 " in obtained binary bit sequence s, it is determined that needing the size of generator matrix chosen；

If bit value is q for the quantity of " 1 " in sequence s, index number h in wherein sequence s_dBit value represent sequence s for 1, d In from a high position to low level occur d-th of 1, d=1,2 ... q；The generator matrix of q polar code is then chosen, respectively size is l₁,l₂,...,l_qSquare formation,D=1,2 ... q, h_dFor positive integer；Sequence s is proceeded by from low level to a high position from 1 Index number；

Step 3：Selected generator matrix is combined by Combo-Sum computings, the generator matrix that code length is N is obtained；

If obtaining q generator matrix according to step 2Then code length is expressed as N generator matrix：

Wherein,Represent Combo-Sum computings；

Described Combo-Sum computings are：If the matrix for participating in computing isWithObtaining combinatorial matrix C is：

$C_{(L_1+L_2)\×(L_1+L_2)}=\left[\begin{array}{cc}A& 0\\ B^{\′}& B\end{array}\right]$

Wherein, L₁、L₂Respectively matrix A, B size, matrix B ' size are L₂×L₁, the preceding L in B '₂Row are identical with matrix B, after L₁-L₂It is classified as full 0 row.

2. the variable Polar code code construction methods of a kind of code length according to claim 1, it is characterised in that described In step 3, described Combo-Sum computings, implementation method is：If two matrixes for participating in Combo-Sum computings areWithThe Combo-Sum operation results of two matrixes areIts Middle element

$c_{ij}=\left\{\begin{array}{cc}{a}_{ij},& i,j\≤L_1\\ b_{i-L_1,j-L_1},& i,j>L_1\\ b_{i-L_1,j_0},& i>L_1,j\∈\mathrm{\Γ}\\ 0,& others\end{array}\right.$

Wherein, Γ is { 1,2 ..., L₁A subset, referred to as difference collection, set Γ element number is L₂, rope in set Γ Tendering is designated as j₀=1,2 ..., l₂, set Γ jth₀Individual element value be C in index j.

3. the variable Polar code code construction methods of a kind of code length according to claim 2, it is characterised in that described In Combo-Sum computings, the A of computing is participated in, B matrixes are unable to exchange sequence, and the size of first matrix A can not be less than second Matrix B.