CN106685432A - Large girth type-II QC-LDPC code construction method based on complete cyclic difference sets - Google Patents
Large girth type-II QC-LDPC code construction method based on complete cyclic difference sets Download PDFInfo
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- H—ELECTRICITY
- H03—ELECTRONIC CIRCUITRY
- H03M—CODING; DECODING; CODE CONVERSION IN GENERAL
- H03M13/00—Coding, decoding or code conversion, for error detection or error correction; Coding theory basic assumptions; Coding bounds; Error probability evaluation methods; Channel models; Simulation or testing of codes
- H03M13/03—Error detection or forward error correction by redundancy in data representation, i.e. code words containing more digits than the source words
- H03M13/05—Error detection or forward error correction by redundancy in data representation, i.e. code words containing more digits than the source words using block codes, i.e. a predetermined number of check bits joined to a predetermined number of information bits
- H03M13/11—Error detection or forward error correction by redundancy in data representation, i.e. code words containing more digits than the source words using block codes, i.e. a predetermined number of check bits joined to a predetermined number of information bits using multiple parity bits
- H03M13/1102—Codes on graphs and decoding on graphs, e.g. low-density parity check [LDPC] codes
- H03M13/1148—Structural properties of the code parity-check or generator matrix
- H03M13/116—Quasi-cyclic LDPC [QC-LDPC] codes, i.e. the parity-check matrix being composed of permutation or circulant sub-matrices
- H03M13/1168—Quasi-cyclic LDPC [QC-LDPC] codes, i.e. the parity-check matrix being composed of permutation or circulant sub-matrices wherein the sub-matrices have column and row weights greater than one, e.g. multi-diagonal sub-matrices
Abstract
The invention relates to a large girth type-II QC-LDPC (Quasi Cyclic-Low Density Parity Check) code construction method based on complete cyclic difference sets. The large girth type-II QC-LDPC code construction method based on complete cyclic difference sets aiming at the problem that as Weight-2Circulant Matrices (W2CM) exist in verification matrices of current type-II QC-LDPC codes, a Tanner graph is easier to generate a short girth, so that iteration decoding convergence is influenced, and fully utilizes the special characteristics of complete cyclic difference sets (CDS) to use the complete CDS to construct verification matrices of the type-II QC-LDPC codes so as to avoid generation of short girth. The large girth type-II QC-LDPC code construction method based on complete cyclic difference sets includes the steps: constructing a shift matrix S(H) with a central symmetrical structure, wherein two groups of element pairs which are formed by the elements concentrated in different complete cyclic difference sets exist on the leading diagonal of the S(H), and the elements 0 and infinite are on the other positions; and utilizing a zero matrix, a unit matrix and the W2CM to expand the S(H) to a verification matrix H, wherein the W2CM matrix exists on the leading diagonal of the H. Through theoretical derivation, the large girth type-II QC-LDPC code construction method based on complete cyclic difference sets guarantees that 4-girth and 6-girth do not exist in the H, so that the girth of the constructed type-II QC-LDPC codes is 8, and also verifies that the code words constructed by the large girth type-II QC-LDPC code construction method based on complete cyclic difference sets have excellent performance through simulated verification.
Description
Technical field
The invention belongs to field of signal processing, is related to chnnel coding, a kind of especially enclosing greatly based on complete circular difference set
Long type-II QC-LDPC code constructing methods.
Background technology
Digital communication industry is that one huge and the industry of fast development, and its scope covers wireless mobile communications, satellite
The fields such as the data communication between digital broadcast television, computer digit storage system and computer.With believing towards digitized
The transmission of breath, process and emerging in large numbers for mass rapid data network, the demand of high efficient and reliable transmission and storage to information such as store
Grow with each passing day, thus how to reduce mistake with guarantee information can transmitting be digital communication design solve key
One of problem.Error-Control Coding is finding and correct by adding a small amount of redundancy check information in effective information
A kind of error code produced in transmitting procedure, it is ensured that effective means of information transfer reliability.Low-density checksum (Low-
Density Parity-Check, LDPC) code is a kind of linear block codes with sparse check matrix, its performance is extremely approached
Shannon is limited, and has been widely used in error control coding field.Quasi-circulating low-density parity check (Quasi-Cyclic
Low-Density Parity-Check, QC-LDPC) code is a kind of structural type LDPC code, its check matrix has quasi- circulation special
Property, can realize encoding by simple linear shift register, reduce required memory space, reduce hard-wired complexity
Degree, it has also become the research focus on coding circle.
Generally, the check matrix H of QC-LDPC is made up of circulation submatrix.When in check matrix H only include cyclic permutation
Matrix (Circulant Permutation Matrices, CPM) or null matrix (Zero Matrices, ZM) both forms
Circulation submatrix when, corresponding code word is type-I QC-LDPC codes, and the at present QC-LDPC codes constructed by most of documents are all
It is belonging to type-I QC-LDPC codes.And the circulation that type-II QC-LDPC code check matrixes H includes CMP, ZM and weight is 2
Matrix (Weight-2Circulant Matrices, W2CM), with type-I QC-LDPC code-phase ratios, it has bigger minimum
Apart from upper dividing value, minimum range is directly related with code error-correcting performance, and lowest distance value is bigger, and the anti-interference of code is better, error detection
Error correcting capability is also stronger.When middle short code is long, the performance of type-II QC-LDPC codes is even better than the LDPC code of random configuration.
However, presence of the W2CM in check matrix is easier becate occur in will making Tanner figures, when adopting iterative decoding algorithm
When, by convergence rate during impact iterative decoding, so as to cause decoding performance to decline.Jing consulting literatures understand, presently, there are
It is only there is no Fourth Ring, but the presence of a large amount of six rings to decoding convergence in the check matrix H of type-II QC-LDPC codes
There is certain impact.Complete circular difference set (Cyclic Difference Sets, CDS) is that a class is highly important in Combinational Mathematics
Design theory, it is also different that its main character is the difference of any two difference element in set, and complete CDS is incorporated into
The generation of becate can be prevented effectively from the construction of type-II QC-LDPC codes.Therefore, the present invention proposes one using complete CDS
The building method of the type-II QC-LDPC codes for enclosing a length of 8 is planted, the generation of Fourth Ring and six rings is efficiently avoid, so as to construct
The code word of excellent performance is gone out.
The content of the invention
In view of this, it is an object of the invention to provide a kind of enclose greatly long type-II QC- based on complete circular difference set
LDPC code building method, by designing a kind of check matrix that there are no four or six rings, so as to construct the code word of excellent performance.
To reach above-mentioned purpose, the present invention provides following technical scheme：
It is a kind of that long type-II QC-LDPC code constructing methods are enclosed greatly based on complete circular difference set, including：
1., for convenience of narration technical scheme, the structure of type-II QC-LDPC codes is first introduced
If L, p are two positive integers, form of the code length for the check matrix H of the type-II QC-LDPC codes of N=Lp
As shown in (1).
Wherein, J≤L, to any 0≤j≤J-1,0≤l≤L-1, i ∈ { 1,2 },Represent unit square
The number of times of the often capable cyclic shift to the right of battle array.WhenWhen, I (∞) represents p × p null matrix 0, whenWhen, I
(0) it is p × p unit matrix Ip, whenWhen,Represent that the unit matrix of a p × p is often capable to move rightPosition
The CPM of gained.From formula (1) as can be seen that check matrix H includes following three form：The CPM that ZM that weight is 0, weight are 1
With the circular matrix that weight is 2WhereinOne type-II QC-LDPC code can also use check matrix
The shift matrix S (H) of H is represented.
2. the structure according to type-II QC-LDPC codes, after its shift matrix S (H) determines, corresponding check matrix
Determine therewith.The present invention constructs the shift matrix S (H) of 4 × 8 first, as shown in Figure 2.The moon on the leading diagonal of S (H)
Shadow frame SAi(0≤i≤3)SAi(0≤i≤3) are that two constituent element elements are right, take one group of (v, k, 1) complete circular difference set D={ d1,
d2,...,dk, k >=16, in each dash box, the value successively of element pair is two element (d for differing in Du,dc), 1≤u
＜ c≤k, in D, each element only occurs once, makes the interval that δ=c-u is every constituent element element two element of centering, if k is even number,
δ=k/2 is taken then, k then takes δ=(k+1)/2 for odd number；It is 0 element on the position of dotted line frame；It is ∞ on remaining position；
3. spreading factor p=v made, and the element in S (H) is to circular matrix I (d that the weight for replacing with a p × p is 2u)
+I(dc), each I (di) (1≤i≤k) represent unit matrix cyclic shift d to the rightiCPM after secondary；0 element replaces with a p
The unit matrix I of × pp；∞ replaces with p × p null matrix 0.According to the structure of code, check matrix H also has center
Symmetrical structure.
The beneficial effects of the present invention is：The method construction type-II QC-LDPC codes check matrix by null matrix,
Unit matrix and W2CM compositions, remain type-II QC-LDPC codes has the advantages that the higher minimum range upper bound, improves
The error-correcting performance of code；And the appearance without four or six rings in check matrix, there is during decoding convergence rate faster.Simulation result table
Bright, the type-II QC-LDPC codes for enclosing a length of 8 for being constructed are believed at (Additive white Gaussian noise, AWGN)
When being decoded using confidence spread (Belief Propagation, BP) algorithm iteration under road, under equal conditions, its error correction
The PEG codes of random method construction can be better than, a length of 6 type-II QC-LDPC codes are enclosed and enclosed a length of 8 type-I QC-LDPC
Code.
Description of the drawings
In order that the purpose of the present invention, technical scheme and beneficial effect are clearer, the present invention provides drawings described below and carries out
Explanation：
Technology Roadmaps of the Fig. 1 for the inventive method；
Fig. 2 is the structure of the shift matrix S (H) of present invention construction；
The form that Fig. 3 is present for six rings in the shift matrix S (H) of type-II QC-LDPC codes；
Fig. 4 is girth-8 type-II CDS QC-LDPC (2184,1092) codes that the code check of the construction of embodiment 1 is 0.5
With the Performance comparision figure of other yards；
Fig. 5 is girth-8 type-II CDS QC-LDPC (6056,3028) codes that the code check of the construction of embodiment 2 is 0.5
With the Performance comparision figure of other yards.
Specific embodiment
Below in conjunction with accompanying drawing, the preferred embodiments of the present invention are described in detail.
1. combine accompanying drawing 1 to illustrate, for module Z of a v rankv={ 0,1,2 ..., v-1 }, D={ d1,d2,...,dk}
For ZvThe subset containing k element, ZvIn each nonzero element in (di-dj) modv computings result in there is just λ
Secondary, then D is referred to as Zv(v, k, a λ) circular difference set.The parameter that circular difference set can be obtained by definition meets：λ=k (k-1)/(v-
1).When λ=1, v=k2- k+1, such circular difference set is called complete circular difference set, any two in complete circular difference set D
Element di, djMould v difference set (di-dj) modv operation results are different.For any prime number power q=pm, p be a prime number, m
For arbitrary positive integer, v=q2+ q+1, for moduleThere is (a q2+ q+1, q+1,1) complete circular difference set.
The complete circular difference set in part is listed in table 1.
1 part of table (v, k, 1) complete circular difference set (4≤k≤30)
2. combine accompanying drawing 1 to illustrate, construct the shift matrix S (H) of 4 × 8, as shown in Figure 2.On the leading diagonal of S (H)
Dash box SAi(0≤i≤3)SAi(0≤i≤3) are that two constituent element elements are right, take one group of (v, k, 1) complete circular difference set D={ d1,
d2,...,dk, k >=16, in each dash box, the value successively of element pair is two element (d for differing in Du,dc), 1≤u
＜ c≤k, in D, each element only occurs once, makes the interval that δ=c-u is every constituent element element two element of centering, if k is even number,
δ=k/2 is taken then, k then takes δ=(k+1)/2 for odd number；It is 0 element on the position of dotted line frame；It is ∞ on remaining position；
3. combine accompanying drawing 1 to illustrate, make spreading factor p=v, the element in S (H) is 2 to the weight for replacing with a p × p
Circular matrix I (du)+I(dc), each I (di) (1≤i≤k) represent unit matrix cyclic shift d to the rightiCPM after secondary；0
Element replaces with the unit matrix I of a p × pp；∞ replaces with p × p null matrix 0.According to the structure of code, verification
Matrix H also has centrosymmetric structure.
4. combine accompanying drawing 1 to illustrate, the method for proof without Fourth Ring in H
For all j0,j1, 0≤j0≠j1≤ J-1, all l0,l1, 0≤l0≠l1≤ L-1, and all it∈ { 1,2 },
0≤t≤3, and if only if when following each equation is false, corresponding to the check matrix H of type-II QC-LDPC codes
There is no Fourth Ring in Tanner figures.
①
②And
③And
④
Wherein i0,i1∈ { 1,2 }, 0≤j0,j1≤ J-1,0≤l0,l1≤ L-1, it should be noted that work as j0=j1And l0=l1When,
i0≠i1。
According to more than 4 equatioies, in shift matrix S (H) in addition to ∞ the different element of any twoWithIt
Between difference under mod p computings it is different when, this four equatioies are all false.Due to any two in complete circular difference set D
Individual element du,dcMould v difference set (dc-du) modv operation results are different, therefore the method for construction of the invention, S
(H) it is not the presence of Fourth Ring in, then in corresponding check matrix H, also there is no Fourth Ring.
5. combine accompanying drawing 1 to illustrate, the method for proof without six rings in H
The sufficient and necessary condition that there are no 6 rings in H can be written as：
With reference to the structure of S in Fig. 2 (H), there are no six rings from the H of following several situation analysis explanation present invention constructions：
1. work as j0=j1When (give tacit consent to other coordinate variables unequal, below several situations be similar to), (1) formula can be write asConstitute position such as Fig. 3 (a) of six rings.This is understood according to Fig. 2
There is no such form in the shift matrix of invention construction, therefore the situation is false.
J can be obtained in the same manner0=j2And j1=j2Situation it is 1. similar with situation.
2. work as l0=l1When, (1) formula can be write asSix are constituted
The position of ring such as Fig. 3 (b).There is no such form, therefore the situation in the shift matrix that present invention construction is understood according to Fig. 2
It is false.
L can be obtained in the same manner0=l2And l1=l2Situation it is 2. similar with situation.
3. work as j0=j1,l0=l1When, (1) formula can be write as
As, in addition to first and last position, in a ring, all positions all differ, therefore 1. situation is false.
J can be obtained in the same manner0=j1,l0=l2；Or j0=j1,l1=l2；Or j0=j2,l0=l1；Or j1=j2,l0=l1Feelings
Condition is 3. similar with situation.
4. work as j0=j1=j2When, (1) formula can be write asCorrespondence
The position of six rings such as Fig. 3 (c).There are no such form, therefore the feelings in the shift matrix that present invention construction is understood according to Fig. 2
Condition is false.
5. work as l0=l1=l2When, (1) formula can be write asThat is structure
The position of Cheng Liuhuan such as Fig. 3 (d).There is no such form in the shift matrix that present invention construction is understood according to Fig. 2, therefore should
Situation is false.
6. work as j0≠j1≠j2, l0≠l1≠l2When, be equivalent to the form that six rings are present in type-I QC-LDPC codes, i.e., six
Ring is necessarily arranged across three rows three of S (H), it is clear that the situation is also false.
In sum, there are no six rings in the check matrix H of present invention construction.
6. error bit ability analysis
Two embodiments are presented below illustrate and analyze the present invention constructed enclose a length of 8 type-II QC-
The performance of LDPC code.Simulated environment uses binary phase shift keying (BPSK) modulation under awgn channel, confidence spread
(BP) algorithm iteration decoding, maximum iteration time take 50 times.And the PEG codes with random configuration, enclose a length of 6 type-II QC-
LDPC code carries out performance comparison with the type-I QC-LDPC codes for enclosing a length of 8.
Embodiment 1：Take (273,17,1)-complete circular difference set D=1,2,23,34,84,123,135,142,146,
160,176,201,226,230,232,239,247 }, p=v=273, δ=9 are made, it is 8 that construction one kind is enclosed and grows (girth), code check
For 0.5, code length is 2184 type-II QC-LDPC codes, be designated as girth-8 type-II CDS-QC-LDPC (2184,
1092) code.And with constructed based on complete CDS enclose a length of 6 type-II CDS-QC-LDPC (2212,1108) code, based on most
What big common divisor (Greatest Common Divisor, GCD) was constructed enclose a length of 8 type-I GCD-QC-LDPC (2200,
1100) code and random configuration PEG (2184,1092) code carry out Performance comparision, simulation result and data are respectively such as Fig. 4 and Biao
Shown in 2.From Fig. 4 and Biao 2 as can be seen that when bit error rate (Bit Error Rate, BER) is 10-5When, what the present invention was constructed
Girth-8 type-II CDS-QC-LDPC (2184,1092) code and PEG (2184,1092) code-phase is slightly better than performance, than
Girth-6 type-II CDS-QC-LDPC (2212,1108) codes girth-8 type-I GCD-QC-LDPC (2200,1100)
The net coding gain (Net Coding Gain, NCG) of code improves 0.38dB and 0.11dB respectively.
The pattern constructed in 2 embodiment 1 of table and the Performance comparision of other patterns
| Pattern | Code length | Information bit | Code check | BER=10-5NCG |
| Girth-8 type-II CDS-QC-LDPC codes | 2184 | 1092 | 0.5 | 7.49dB |
| PEG codes | 2184 | 2184 | 0.5 | 7.42dB |
| Girth-8 type-I GCD-QC-LDPC codes | 2200 | 1100 | 0.5 | 7.38dB |
| Girth-6 type-II CDS-QC-LDPC codes | 2212 | 1108 | 0.5 | 7.11dB |
Embodiment 2：Take (757,28,1)-complete circular difference set D=1,2,63,103,112,114,119,158,171,
199,242,264,333,345,363,371,405,408,437,556,591,644,661,680,711,734,738,744 },
Make p=v=757, δ=14, construction one kind are enclosed for 8, and code check is 0.5, and code length is 6056 type-II QC-LDPC codes, is designated as
Girth-8 type-II CDS-QC-LDPC (6056,3028) codes.And with enclose a length of 6 based on Sidon sequence structures
Type-II Sidon-QC-LDPC (6056,3028) codes and based on GCD construct enclose a length of 8 type-I GCD-QC-LDPC
(6100,3050) code carry out Performance comparision, simulation result and data are respectively as shown in Fig. 5 and Biao 3.Can from Fig. 5 and Biao 3
Go out, when BER is 10-5When, the present invention constructed girth-8 type-II CDS-QC-LDPC (6056,3028) code ratio
Girth-6 type-II Sidon-QC-LDPC (6056,3028) codes and girth-8 type-I GCD-QC-LDPC (6100,
3050) NCG of code improves 0.34dB and 0.12dB respectively.
The pattern constructed in 3 embodiment 2 of table and the Performance comparision of other patterns
| Pattern | Code length | Information bit | Code check | BER=10-5NCG |
| Girth-8 type-II CDS-QC-LDPC codes | 6056 | 3028 | 0.5 | 7.88dB |
| Girth-8 type-I GCD-QC-LDPC codes | 6100 | 3050 | 0.5 | 7.76dB |
| Girth-6 type-II Sidon-QC-LDPC codes | 6056 | 3028 | 0.5 | 7.54dB |
Finally illustrate, preferred embodiment above is only unrestricted to illustrate technical scheme, although logical
Cross above preferred embodiment to be described in detail the present invention, it is to be understood by those skilled in the art that can be
Various changes are made to which in form and in details, without departing from claims of the present invention limited range.
Claims (3)
- It is 1. a kind of that long type-II QC-LDPC code constructing methods are enclosed greatly based on complete circular difference set, it is characterised in that：For working as Front type-II low-density checksums (Quasi-Cyclic Low-Density Parity-parity check, QC-LDPC) There is the circular matrix (Weight-2Circulant Matrices, W2CM) that weight is 2 in the check matrix of code causes Tanner figures are more prone to becate, so as to affect the constringent problem of iterative decoding, make full use of complete circular difference set The special nature of (Cyclic Difference Sets, CDS), complete CDS is used for construct the school of type-II QC-LDPC codes Matrix is tested to avoid the generation of becate, a shift matrix S (H) with centrosymmetric structure is constructed first, the master of S (H) is right On linea angulata be by different complete circular difference sets in elementary composition two constituent element elements it is right, be element 0 and ∞ on remaining position, so S (H) is expanded to into check matrix H using null matrix, unit matrix and W2CM afterwards, is W2CM arrays on the leading diagonal of H.Pass through Theoretical derivation is demonstrated during the building method ensure that H does not have Fourth Ring and six rings, therefore the type-II QC-LDPC codes for constructing Enclose a length of 8.
- 2. long type-II QC-LDPC code constructing methods are enclosed greatly based on complete circular difference set according to right 1 is required, which is special Levy and be：Shift matrix S (H), with centrosymmetric structure, is by the element group in different complete circular difference sets on leading diagonal Into two constituent element elements it is right, if (v, k, 1) complete circular difference set be D={ d1,d2,...,dk, wherein k >=16, the element centering per constituent element The value successively of element pair is two element (d for differing in Du,dc), 1≤u ＜ c≤k, in D, each element only occurs once, The interval for making δ=c-u be every constituent element element two element of centering, if k is even number, takes δ=k/2, and k then takes δ=(k+ for odd number 1)/2；It is 0 and ∞ on remaining position.
- 3. long type-II QC-LDPC code constructing methods are enclosed greatly based on complete circular difference set according to right 1 is required, which is special Levy and be：By the unit in the unit matrix replacement of the element 0 in shift matrix S (H), the replacement of element ∞ null matrix, complete CDS The element of element composition to being replaced with W2CM after obtain check matrix H.W2CM therein is moved by the circulation of two unit matrixs Bit matrix is added in two element field and obtains, and the value of element centering element is the number of times of unit matrix displacement.It is final by the method There are no Fourth Ring and six rings in the check matrix H of construction, further improve decoding convergence, so as to improve error-correcting performance.
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| CN108270451A (en) * | 2018-01-22 | 2018-07-10 | 西安电子科技大学 | A kind of Enhancement Method of quasi- fluxoid LDPC code applied to quantum communication system |
| CN108270451B (en) * | 2018-01-22 | 2021-06-04 | 西安电子科技大学 | Method for enhancing quasi-cyclic quantum LDPC code applied to quantum communication system |
| CN108768408A (en) * | 2018-05-17 | 2018-11-06 | 西安空间无线电技术研究所 | A kind of big girth II types quasi-cyclic LDPC code design method based on Sidon sequences |
| CN108768408B (en) * | 2018-05-17 | 2022-05-24 | 西安空间无线电技术研究所 | Large girth II type quasi-cyclic LDPC code design method based on Sidon sequence |
| CN109802689A (en) * | 2019-03-13 | 2019-05-24 | 重庆邮电大学 | A kind of QC-LDPC code constructing method that the girth based on Hoey sequence is 8 |
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