CN101252360A - Structured constitution method of high enclose long low code rate multi-scale LDPC code - Google Patents

Abstract

The invention relates to a construction method of structured non-binary LDPC code with large girth and low bit-rate based on finite field. The method includes the following steps: 1, parameters m and s are selected; 2, a subset B of GF(2<ms>) is constructed; 3, a base matrix W is constructed on the GF(2<ms>) by taking use of the subset B; 4, each element of the base matrix W is replaced with the address vector of the element and the nary numbers of the none-zero elements of the address vector are selected according to the nary numbers of the LDPC code to be constructed; in this way the LDPC code can be constructed. The construction method of check matrices provided by the invention is capable of constructing non-binary LDPC codes with any minimum length. Simulation results show that algebraic non-binary LDPC codes constructed through the construction method are obviously improved as compared with Mackay non-binary stochastic LDPC codes with the same parameters.

Description

A kind of structured constitution method of high enclose long low code rate multi-scale LDPC code

Technical field

The present invention relates to the structured constitution method that a kind of height based on finite field encloses the m-ary LDPC sign indicating number of long low code rate, belong to communication channel coding and decoding field, be specifically related to a kind of check matrix building method of multi-system low density parity check code.

Background technology

In general, actual channel is not desirable.At first these channels all have nonideal frequency selective characteristic, other interference that mix up into when also having noise jamming and signal by Channel Transmission in addition.These interference have damaged the transmission signal, and make the Serial No. of reception produce mistake.In order to overcome these noises and interference, increase reliability of data transmission, in other words in order to increase the fidelity of received signal, usually need in information sequence, introduce some redundant digits.This increase data redudancy is called chnnel coding with anti-disturbance method.

Low density parity check code (LDPC) is proposed by Gallager the earliest, and the back causes extensive attention after being rediscovered by Mackay, is the error correcting code of a class excellent performance.T.J.Richardson and R.Urbanke is by discovering, under BPSK modulation, awgn channel, code check is that 1/2 LDPC code distance shannon limit only is 0.0045dB, this also be find since the dawn of human civilization near the code word of shannon limit.

The LDPC sign indicating number is a kind of linear block codes, is gained the name by the sparse property of check matrix, that is: the nonzero element number of check matrix is much smaller than the number of neutral element.Its check matrix can be represented with a Tanner figure (bipartite graph), as shown in Figure 1, is commonly referred to as Tanner figure.There is N node in bottom under the figure, and each node is represented the information bit of code word, is called information node { x _j, j=1,2 ..., N} is also referred to as variable node, is the bit of code word, corresponding to each row of check matrix; There is M node the top of figure, and each node is represented a checksum set of code word, is called check-node { z _i, i=1,2 ..., M} represents check equations, corresponding to each row of check matrix; Element H in check matrix _Ij, between i check-node among the Tanner figure and j variable node line is arranged at=1 o'clock.The limit number that links to each other with each node is called the number of degrees (degree) of this node, and for example, check matrix of (10,2,4) LDPC sign indicating number and Tanner scheme as shown in Figure 1, and the number of degrees of information node are 2, and the number of degrees of check-node are 4.If the degree of each variable node or check-node is identical among the Tanner figure, claim that then corresponding LDPC sign indicating number is regular LDPC sign indicating number, if the degree of each variable node or check-node is incomplete same among the Tanner figure, claim that then corresponding LDPC sign indicating number is an irregular LDPC codes.

Ring among the Tanner figure (cycle) is defined as the summit and the limit replaces a finite sequence of forming, and starting point and terminal point are that same summit and each summit can only occur once.The length of ring is exactly the number on limit wherein.Ring among the Tanner figure is by variable node, check-node and be connected the closed circuit that their limit joins end to end and forms.For the Tanner of LDPC sign indicating number figure, becate length is 4, and long 2 the multiple that is of all rings.As shown in Figure 2, the dotted line among the Tanner figure encircles 4 exactly, the element that indicates with square frame in the corresponding check matrix.

The factor of performance that influences the LDPC sign indicating number is more, and for example becate is long, the degree distribution of node etc.It is independent that the core concept of LDPC sign indicating number iterative decoding algorithm is based on the Information Statistics of transmitting between node, when having ring to exist among the Tanner figure of LDPC code check matrix correspondence, can be passed back itself after the transmission of a ring length of the information via that a certain node sends, thereby cause the stack of self information, the information of having destroyed prior supposition is independent, the performance of influence decoding.Theory and practice all proves, and the ring among the Tanner figure, particularly minimum ring length are very big to the performance impact of iterative decoding.In the check matrix of LDPC sign indicating number structure, especially to avoid the appearance of becate long 4.

The LDPC sign indicating number that Mackay and Neal proposed on the multi-system territory in 1998, and proved than binary system LDPC code performance and improved a lot with Turbo code certain comparativity is arranged, even can surpass Turbo code, as shown in Figure 3, this makes and has reached a higher step in the development of LDPC sign indicating number.The m-ary LDPC sign indicating number is different with binary system LDPC sign indicating number mainly is that the nonzero element of the check matrix H of the LDPC sign indicating number on the GF (q) can have q-1 value, and not only is " 1 ".The m-ary LDPC sign indicating number is after proposing, and its application also is widely studied.Nearest Hongxin Song etc. studies show that the error-correcting performance of m-ary LDPC sign indicating number in disk storage system is better than the performance as the RS sign indicating number of one of the short frame sign indicating number of the high code check of disk storage system industrial standard.The m-ary LDPC sign indicating number also is studied and is used for optical transmission system, is 10 at BER ^-10The time, the redundancy with 12.59% has obtained the coding gain of 9.9dB.

The structure of LDPC sign indicating number is an emphasis of LDPC sign indicating number research.Building method mainly is divided into random configuration method and structure building method.The LDPC sign indicating number that structured constitution method constructs has good architectural characteristic, helps reducing volume, decoding complexity and check matrix memory space.And the algebraically building method is an emphasis of structured configurations LDPC sign indicating number research.For binary system LDPC sign indicating number, people such as Shu Lin have proposed the method construct LDPC sign indicating number based on different algebraic processs such as finite field, finite geometry, Combination Design, the binary system LDPC of these method constructs has with the LDPC sign indicating number is the same excellent at random, even be better than the performance of random code, obtained extensive use.And for the m-ary LDPC sign indicating number, it is relative less that the progress of building method aspect is compared with binary system, but the algebraically building method remains an emphasis of multi-system structured constitution method.People have also proposed based on finite field, the method for finite geometry scheduling theory, and comparing with the RS sign indicating number with the m-ary LDPC sign indicating number of these algebraic methods structure has bigger gain.

But, it is that 4 ring occurs that the algebraically building method of present existing m-ary LDPC sign indicating number only is confined to avoid length, only can guarantee that promptly minimum ring length is 6, and forefathers after deliberation m-ary LDPC code book body just superior than RS code performance, therefore only the m-ary LDPC sign indicating number of algebraically structure is compared the method that can not embody algebraically structure m-ary LDPC sign indicating number and the superiority of other structures LDPC code method with the RS sign indicating number.At two top problems, the building method of the check matrix that the present invention proposes can construct the long m-ary LDPC sign indicating number of minimum ring arbitrarily, and in emulation of the present invention, the sign indicating number that contrasts all adopts the sign indicating number of LDPC at random of Mackay, with outstanding superiority as algebraic method of the present invention.

Summary of the invention

The present invention grows and sends out from increasing minimum ring on the basis of the finite field algebraic method of existing m-ary LDPC code check matrix, has proposed a kind of method based on finite field structure high enclose long low code rate multi-scale structured LDPC code.The sharpest edges that this method is compared with existing m-ary LDPC code constructing method are exactly to select to construct the long m-ary LDPC sign indicating number of high arbitrarily minimum ring by parameter.

This method has been utilized this basic principle of element that necessarily has linear independence in the linear space to make the LDPC code check matrix construct have height to enclose length, and concrete principle is proved in the back.

Technical scheme of the present invention is as follows: a kind of method of constructing the high enclose long low code rate multi-scale structured LDPC code, and its feature comprises following concrete steps:

The first step is selected two positive integer parameter m, s; Make q=2 ^Ms

Second step is with finite field gf (2 ^Ms) be considered as GF (2 ^s) on linear space, construct finite field gf (2 ^Ms) a subclass B={b that element number is m+1 ₁, b ₂..., b _M+1, make any m element linear independence all among the B; Such set B is not only, and wherein a kind of common constitution method is B={1, α, α ²..., α ^M-1, 1+ α+... α ^M-1.

The 3rd step was utilized set B structure GF (2 ^Ms) on basic matrix W; Suppose that α is GF (2 ^Ms) a primitive element, for i=1,2 ..., m+1 constructs q * 2 respectively ^sSubmatrix W _i

α wherein ^-∞=0, α ⁰=1, α ..., α ^Q-2Be GF (2 ^Ms) in all elements, β ⁰, β ¹..., β ^2s-2Be GF (2 ^s) in all nonzero elements.M+1 submatrix formed a line, obtain mq * 2 ^sBasic matrix W

$W=\left[\begin{array}{c}{W}_1\\ {\mathrm{<figure-callout\; id="2"\; label="W"\; filenames="S2008101025049E00013.png,S2008101025049E00021.png"\; state="\{\{state\}\}">W</figure-callout>}}_2\\ .\\ .\\ .\\ W_{m+1}\end{array}\right]$

The 4th step substituted each element among the basic matrix W with its address vector, and selected the system number of address vector nonzero element according to the system number of the LDPC sign indicating number of required structure, promptly got the check matrix H of the LDPC sign indicating number of constructing to the end, was of a size of mq * 2 ^s(q-1); So-called address vector just is meant any one the element α among the GF (q) ⁱ(0≤i＜q-1) all with the vectorial z (α of a q-1 dimension ⁱ)=(z ₀, z ₁..., z _Q-2) correspondence, this vector has only element z _iBe nonzero element, all the other elements all are 0, and unique nonzero element z _iCan be α ⁱItself also can be the element of the finite field on other rank, by the system number decision of required structure LDPC sign indicating number.

When being the selection of set B in second step, the key of this method utilized the structure of linear independence between some elements and the 3rd step basic matrix.The minimum ring length of the last check matrix H that this method constructs is at least 2m+2, below this conclusion of proof.

Proof: prove that at first not having length among the Tanner figure of check matrix is the ring of 2m.Supposing to exist among the Tanner figure length is the ring of 2m, and necessarily having m so in basic matrix W, capable (note is made u ₁, u ₂..., u _m∈ [1, (m+1) q]) and m row (note is made x ₁, x ₂..., x _m∈ [1,2 ^s]) make u ₁Row and u ₂Row is at a certain row x ₁Corresponding element equate u ₂Row and u ₃Row is at a certain row x ₂Corresponding element equate ..., u _M-1Row and u _mRow is at a certain row x _M-1Corresponding element equate u _mRow and u ₁Row is at a certain row x _mCorresponding element equate.On the other hand, according to each submatrix W of basic matrix _iStructure as can be known, each element of the arbitrary row in each submatrix all is inequality, so this u ₁, u ₂..., u _mAny two provisional capitals in the row can be at same submatrix W _iIn.Be without loss of generality, might as well establish u ₁Row is at W ₁In, u ₂Row is at W ₂In ..., u _mRow is at W _mIn, we can obtain following equation group so:

$\left\{\begin{array}{c}{\mathrm{\&alpha;}}^{i_1}+{\mathrm{\&beta;}}^{j_1}{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="S2008101025049E00012.png,S2008101025049E00013.png"\; state="\{\{state\}\}">b</figure-callout>}}_1={\mathrm{\&alpha;}}^{i_2}+{\mathrm{\&beta;}}^{j_1}{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="S2008101025049E00013.png,S2008101025049E00021.png"\; state="\{\{state\}\}">b</figure-callout>}}_2\\ {\mathrm{\&alpha;}}^{i_2}+{\mathrm{\&beta;}}^{j_2}{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="S2008101025049E00013.png,S2008101025049E00021.png"\; state="\{\{state\}\}">b</figure-callout>}}_2={\mathrm{\&alpha;}}^{i_3}+{\mathrm{\&beta;}}^{j_2}{\mathrm{<figure-callout\; id="3"\; label="b"\; filenames="S2008101025049E00021.png,S2008101025049E00022.png"\; state="\{\{state\}\}">b</figure-callout>}}_3\\ .\\ .\\ .\\ {\mathrm{\&alpha;}}^{i_{m-1}}+{\mathrm{\&beta;}}^{j_{m-1}}{b}_{m-1}={\mathrm{\&alpha;}}^{i_m}+{\mathrm{\&beta;}}^{j_{m-1}}{b}_m\\ {\mathrm{\&alpha;}}^{i_m}+{\mathrm{\&beta;}}^{j_m}{b}_m={\mathrm{\&alpha;}}^{i_1}+{\mathrm{\&beta;}}^{{\mathrm{<figure-callout\; id="1"\; label="j\; m\; b"\; filenames="S2008101025049E00012.png,S2008101025049E00013.png"\; state="\{\{state\}\}">j</figure-callout>}}_m}{b}_{}{}_1\end{array}\right.$

β wherein ^Ji(∞＜j _i＜2 ^s-1) expression vector (0,1, β ..., β ^2s-2) x _iThe element of individual position (1≤i≤m).This m equation group phase adduction transposition can be got:

$b_1({\mathrm{\&beta;}}^{j_1}-{\mathrm{\&beta;}}^{j_m})+b_2({\mathrm{\&beta;}}^{j_2}-{\mathrm{\&beta;}}^{j_1})+\&CenterDot;\&CenterDot;\&CenterDot;+b_m({\mathrm{\&beta;}}^{j_m}-{\mathrm{\&beta;}}^{j_{m-1}})=0$

Because b ₁, b ₂..., b _mBe m element in the set B, their linear independences can get thus ${\mathrm{\&beta;}}^{{\mathrm{<figure-callout\; id="1"\; label="j"\; filenames="S2008101025049E00012.png,S2008101025049E00013.png"\; state="\{\{state\}\}">j</figure-callout>}}_1}={\mathrm{\&beta;}}^{{\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="S2008101025049E00013.png,S2008101025049E00021.png"\; state="\{\{state\}\}">j</figure-callout>}}_2}=\&CenterDot;\&CenterDot;\&CenterDot;={\mathrm{\&beta;}}^{j_m}.$ And vector (0,1, β ..., β ^2s-2) each element just in time constitute GF (2 ^s) in 2 ^sIndividual element, therefore ${\mathrm{\&beta;}}^{{\mathrm{<figure-callout\; id="1"\; label="j"\; filenames="S2008101025049E00012.png,S2008101025049E00013.png"\; state="\{\{state\}\}">j</figure-callout>}}_1}={\mathrm{\&beta;}}^{{\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="S2008101025049E00013.png,S2008101025049E00021.png"\; state="\{\{state\}\}">j</figure-callout>}}_2}=\&CenterDot;\&CenterDot;\&CenterDot;={\mathrm{\&beta;}}^{j_m}$ This result and β ^J1, β ^J2..., β ^JmBe the vector (0,1, β ..., β ^2s-2) in this fact of element of m diverse location contradict.This has just proved that not having length among the Tanner figure of check matrix is the ring of 2m.Equally, because any m element in set B linear independence all, so (the individual element of l＜m) is linear independence also, and in like manner can prove among the Tanner figure of check matrix does not have the ring of length less than 2m for any l.Therefore we as can be known the minimum ring length of the Tanner figure of check matrix be at least 2m+2.

Description of drawings:

Fig. 1 is that regular LDPC code check matrix and Tanner figure give an example.

Fig. 2 is the ring among check matrix and the Tanner figure.

Fig. 3 is that m-ary LDPC sign indicating number and binary system LDPC code performance compare.

Fig. 4 is that the minimum ring length of the inventive method structure is 8 64 systems (252,77) sign indicating number and corresponding Mackay random code performance comparison.

Fig. 5 is that the minimum ring length of the inventive method structure is 10 16 systems (1020,239) sign indicating number and corresponding Mackay random code performance comparison.

Embodiment:

Below in conjunction with two examples application of the present invention is described in detail, and provided simulation result.

Example 1

The first step is selected m=3, s=2, then q=2 ⁶=64;

The second step construction set B={1, α, α ², 1+ α+α ²;

The 3rd step structure GF (2 ⁶) on 256 * 4 basic matrix;

The 4th step substituted each element in the basic matrix with its address vector, the nonzero element of address vector just adopts element original in the basic matrix, obtains GF (2 ⁶) on 256 * 252 last check matrix.Though this check matrix line number is also more than columns, order has only 175, so the LDPC sign indicating number of this check matrix correspondence is (252,77) sign indicating number of 64 systems, and code check is 0.3056, and column weight is 4, and minimum ring length is 8.

Provided the performance simulation result of this sign indicating number among Fig. 4,, in Fig. 4, also provided and have same code rate in order to contrast, code length, the Mackay of system number and column weight is the performance simulation of LDPC sign indicating number at random.Have simulation result as can be seen, the minimum ring length of (252,77) 64 systems of the inventive method structure is that the performance of the random code of 8 LDPC sign indicating number and corresponding Mackay is compared the gain that 1.5dB is arranged.

Example 2

The first step is selected m=4, s=2, then q=2 ⁸=256

The second step construction set B={1, α, α ², 1+ α+α ², 1+ α+α ²+ α ³}

The 3rd step structure GF (2 ⁸) on 1280 * 4 basic matrix

The 4th step substituted each element in the basic matrix with its address vector, the nonzero element of address vector adopts GF (2 ⁴)={ γ ^-∞, γ ⁰, γ ¹..., γ ¹⁴In non-0 element.Concrete replacement method is as follows: the corresponding γ in first position in the address vector ⁰If (promptly non-0 element in the address vector is in first position, and just this non-0 element is set to γ ⁰, below similar), the corresponding γ in second position ¹..., the 15th the corresponding γ in position ¹⁴, the 16th the corresponding again γ in position ⁰The rest may be inferred.Behind the element in the address vector replacement basic matrix, obtain GF (2 ⁴) on 1280 * 1024 last check matrix.Though this check matrix line number is also more than columns, order has only 781, so the LDPC sign indicating number of this check matrix correspondence is (1020,239) sign indicating number of 16 systems, and code check is 0.2343, and column weight is 5, and minimum ring length is 10.

Provided the performance simulation result of this sign indicating number among Fig. 5,, in Fig. 5, also provided and have same code rate in order to contrast, code length, the Mackay of system number and column weight is the performance simulation of LDPC sign indicating number at random.By simulation result as can be seen, the minimum ring length of the inventive method structure is that the performance of the random code of 10 (1020,239) 16 system LDPC sign indicating numbers and corresponding Mackay is compared the gain of 2dB is nearly arranged.

Claims (5)

1, a kind of height based on finite field encloses the structured constitution method of the m-ary LDPC sign indicating number of long low code rate, it is characterized in that, may further comprise the steps:

The first step is selected two parameter m of basic matrix, and s makes q=2 ^Ms

Second step is with finite field gf (2 ^Ms) be considered as GF (2 ^s) on linear space, construct finite field gf (2 ^Ms) a subclass B={b that element number is m+1 ₁, B ₂..., B _M+1;

The 3rd step was utilized set B structure GF (2 ^Ms) on basic matrix W;

The 4th step substituted each element among the basic matrix W with its address vector, and selected the system number of address vector nonzero element according to the system number of the LDPC sign indicating number of required structure, promptly got the LDPC sign indicating number of constructing to the end.

2, a kind of height based on finite field as claimed in claim 1 encloses the structured constitution method of the m-ary LDPC sign indicating number of long low code rate, it is characterized in that: any m element among the described subclass B of second step be linear independence all.

3, a kind of height based on finite field as claimed in claim 1 encloses the structured constitution method of the m-ary LDPC sign indicating number of long low code rate, it is characterized in that, the building method of basic matrix W is described in the 3rd step:

Suppose that α is GF (2 ^Ms) a primitive element, for i=1,2 ..., m+1 constructs q * 2 respectively ^sSubmatrix W _i

α wherein ^-∞=0, α ⁰=1, α ..., α ^Q-2Be GF (2 ^Ms) in all elements, β ⁰, β ¹..., β ^2s-2Be GF (2 ^s) in all nonzero elements; M+1 submatrix formed a line, obtain mq * 2 ^sBasic matrix W.

$W=\left[\begin{array}{c}{W}_1\\ W_2\\ .\\ .\\ .\\ W_{m+1}\end{array}\right]$

4, a kind of height based on finite field as claimed in claim 1 encloses the structured constitution method of the m-ary LDPC sign indicating number of long low code rate, it is characterized in that: described address vector of the 4th step is meant any one the element α among the GF (q) ⁱ(0≤i＜q-1) all with the vectorial z (α of a q-1 dimension ⁱ)=(z ₀, z ₁..., z _Q-2) correspondence, this vector has only element z _iBe nonzero element, all the other elements all are 0, and unique nonzero element z _iCan be α ⁱItself also can be the element of the finite field on other rank, by the system number decision of required structure LDPC sign indicating number.

5, a kind of height based on finite field as claimed in claim 4 encloses the structured constitution method of the m-ary LDPC sign indicating number of long low code rate, it is characterized in that: the nonzero element of described check matrix not necessarily will be from the finite field gf (2 of basic matrix correspondence ^Ms) in choose, can be according to the system number of the LDPC sign indicating number of required structure and from the finite field of correspondence, choose.

Images