(19)
(11)EP 1 598 942 B1

(12)EUROPEAN PATENT SPECIFICATION

(45)Mention of the grant of the patent:
17.08.2011 Bulletin 2011/33

(21)Application number: 04715448.9

(22)Date of filing:  27.02.2004
(51)International Patent Classification (IPC): 
H03M 13/11(2006.01)
(86)International application number:
PCT/JP2004/002399
(87)International publication number:
WO 2004/077680 (10.09.2004 Gazette  2004/37)

(54)

Check matrix generation for irregular LDPC codes with determined code rate

Prüfmatrixerzeugung für irreguläre LDPC Codes bestimmter Coderate

Génération d'une matrice de controle pour des codes LDPC irréguliers avec débit de codage déterminé


(84)Designated Contracting States:
ES FR GB

(30)Priority: 28.02.2003 JP 2003053162

(43)Date of publication of application:
23.11.2005 Bulletin 2005/47

(73)Proprietor: MITSUBISHI ELECTRIC CORPORATION
Chiyoda-ku Tokyo 100-8310 (JP)

(72)Inventor:
  • MATSUMOTO, Wataru, c/o Mitsubishi Denki K. K.
    Chiyoda-ku, Tokyo 100-8310 (JP)

(74)Representative: Nicholls, Michael John 
J.A. Kemp & Co. 14 South Square
Gray's Inn London WC1R 5JJ
Gray's Inn London WC1R 5JJ (GB)


(56)References cited: : 
WO-A-03/056705
WO-A-2004/006444
JP-A- 2003 198 383
WO-A-03/073621
JP-A- 2003 115 768
  
  • MATSUMOTO W ET AL: "Deterministic irregular low-density parity-check codes design scheme" PROC., KISO KYOKAI SOCIETY MEETING, vol. A-6-12, 10 September 2002 (2002-09-10), page 126, XP002903378
  • VASIC B ET AL: "Lattice low-density parity check codes and their application in partial response systems" PROC., IEEE INTERNATIONAL SYMPOSIUM ON INFORMATION THEORY, ISIT 02, LAUSANNE, SWITZERLAND, 30 June 2002 (2002-06-30), page 453, XP010602164 ISBN: 0-7803-7501-7
  • MATSUMOTO W ET AL: "Irregular extended Euclidean geometry low-density parity-check codes" PROC., INTERNATIONAL SYMPOSIUM ON COMMUNICATION SYSTEMS NETWORKS AND DIGITAL SIGNAL PROCESSING, CSNDSP 2002, STAFFORD, UK, 15 July 2002 (2002-07-15), pages 148-151, XP002370884
  • MATSUMOTO W ET AL: "Irregular low-density parity-check code design based on Euclidean geometries" IEICE TRANSACTIONS ON FUNDAMENTALS OF ELECTRONICS, COMMUNICATIONS AND COMPUTER SCIENCES, ENGINEERING SCIENCES SOCIETY, TOKYO, JP, vol. E86-A, no. 7, July 2003 (2003-07), pages 1820-1834, XP001174812 ISSN: 0916-8508
  • MATSUMOTO W ET AL: "Irregular low-density parity-check code design based on integer lattices" PROC., IEEE INTERNATIONAL SYMPOSIUM ON INFORMATION THEORY, ISIT 2003, YOKOHAMA, JAPAN, 29 June 2003 (2003-06-29), page 3, XP010657031 ISBN: 0-7803-7728-1
  • ROSENTHAL J ET AL: "Constructions of regular and irregular LDPC codes using Ramanujan graphs and ideas from margulis" PROC., IEEE INTERNATIONAL SYMPOSIUM ON INFORMATION THEORY, ISIT 2001, WASHINGTON, WA, US, 24 June 2001 (2001-06-24), - 29 June 2001 (2001-06-29) page 4, XP010552621 ISBN: 0-7803-7123-2
  • VONTOBEL P. O. ET AL: "Irregular codes from regular graphs" PROC., IEEE INTERNATIONAL SYMPOSIUM ON INFORMATION THEORY, ISIT 02, LAUSANNE, SWITZERLAND, 30 June 2002 (2002-06-30), - 5 July 2002 (2002-07-05) page 284, XP010601995 ISBN: 0-7803-7501-7
  • CHIANI M ET AL: "Design and performance evalution of some high-rate irregular low-density parity-check codes" PROC., IEEE GLOBAL TELECOMMUNICATIONS CONFERENCE, GLOBECOM 2001, SAN ANTONIO, TX, vol. VOL. 2 OF 6, 25 November 2001 (2001-11-25), - 29 November 2001 (2001-11-29) pages 990-994, XP001099253 ISBN: 0-7803-7206-9
  • ROSENTHAL J. AND VONTOBEL P.O.: "Constructions of LDPC codes using Ramanujan graphs and ideas from Margulis" PROC., 38TH ALLERTON CONFERENCE ON COMMUNICATION, CONTROL AND COMPUTING, ALLERTON HOUSE, MONTICELLO, ILLINOIS, 4 October 2000 (2000-10-04), - 6 October 2000 (2000-10-06) pages 1-10, XP002416895 Retrieved from the Internet: URL:http://www.hpl.hp.com/personal/Pascal_ Vontobel/publications/papers/pvto-00-3.pdf > [retrieved on 2007-01-25]
  • CHUNG S.-Y. ET AL: "Analysis of Sum-Product Decoding of Low-Density Parity-Check Codes Using a Gaussian Approximation" IEEE TRANSACTIONS ON INFORMATION THEORY, vol. 47, no. 2, February 2001 (2001-02), pages 657-670, XP011027864 ISSN: 0018-9448
  • MATSUMOTO, IMAI: 'Ketteironteki hiseisoku LDPC fugo sekkeiho' 2002 NEN THE INSTITUTE OF ELECTRONICS, INFORMATION AND COMMUNICATION ENGINEERS KISO KYOKAI SOCIETY TAIKAI vol. A-6-12, 10 September 2002 - 13 September 2002, page 126, XP002903378
  
Note: Within nine months from the publication of the mention of the grant of the European patent, any person may give notice to the European Patent Office of opposition to the European patent granted. Notice of opposition shall be filed in a written reasoned statement. It shall not be deemed to have been filed until the opposition fee has been paid. (Art. 99(1) European Patent Convention).


Description

TECHNICAL FIELD



[0001] The present invention relates to a method and an apparatus for generating a check matrix for a low-density parity-check (LDPC) code that is applied as an error correcting code, and more particularly, to a method and an apparatus for generating a check matrix capable of searching a definite and characteristic-stabilized check matrix for the LDPC code.

BACKGROUND ART



[0002] A conventional method of generating check matrixes for LDPC codes will be explained below. In a conventional LDPC code encoding/decoding system, a communication apparatus at a sending side has an encoder and a modulator. On the other hand, an apparatus at a receiving side has a demodulator and a decoder. Prior to explanation of the conventional method of generating check matrixes for LDPC codes, flows of encoding and decoding using LDPC codes are explained first.

[0003] At the sending side, the encoder generates a check matrix H according to the conventional method described later. Then, a generator matrix G is obtained based on the following condition.





[0004] The encoder then receives a message (m1 m2 ... mk) of an information length k, and generates a code word C using the generator matrix G.



[0005] The modulator subjects the generated code word C to a digital modulation such as binary phase shift keying (BPSK), quadrature phase shift keying (QPSK), and multi-valued quadrature amplitude modulation (QAM), and sends the modulated signal.

[0006] At the receiving side on the other hand, the demodulator receives the modulated signal via the channel, and subjects it to a digital demodulation such as BPSK, QPSK, and multi-valued QAM. Then the decoder performs an iterative decoding by "sum-product algorithm" with respect to the demodulated result that is LDPC-coded, and outputs an estimated result (corresponding to the original m1 m2 ... mk).

[0007] A conventional method of generating check matrixes for LDPC codes is explained below. As a check matrix for LDPC codes, the following matrix is proposed in R.G. Gallager, "Low-Density Parity Check Codes", M.I.T Press, Cambridge, MA, 1963 (see Fig. 16).

[0008] The matrix shown in Fig. 16 is a binary matrix of "1" and "0", in which a part of "1" is hatched. Other parts are all "0". In this matrix, the number of "1 "s in one row (expressed as a row weight) is equal to 4, and the number of "1"s in one column (expressed as a column weight) is equal to three. All columns and rows have respective uniform weights. Thus, it is generally called a "regular-LDPC code". In the codes in the Non-patent Literature 1, the matrix is separated into three blocks, for example, and the second and the third blocks are subjected to random permutation, as shown in Fig. 16.

[0009] Because the random permutation has no certain rule, it is required to execute a time-consuming search by a computer to find codes with a better characteristic.

[0010] It is proposed in Y. Kou, S. Lin, and M. P. C. Fossorier, "Low Density Parity Check Codes Based on Finite Geometries: A Rediscovery", ISIT 2000, pp. 200, Sorrento, Italy, June 25-30, 2000 a method using Euclidean geometry codes as the LDPC codes that exhibit a relatively stable and satisfactory characteristic and can definitely generate a matrix without the use of the computer search. This method explains the "regular-LDPC code" consisting of regular ensembles.

[0011] The second literature proposes a method of generating a check matrix for LDPC codes using Euclidean geometry codes EG (2, 26) as a kind of finite geometric codes. This method achieves a characteristic that is located closely but 1.45 decibels away from the Shannon limit at an error rate of 10-4. Fig. 17 is a diagram of a configuration of Euclidean geometry codes EG (2, 22), which has a structure of "Regular-LDPC Codes" with row and column weights of 4 and 4, respectively.

[0012] Euclidean geometry codes EG (m, 2s) have a characteristic defined as follows:
Code length: n=22s-1
Redundant bit length: n-k=3s-1
Information length: k=22s-3s
Minimum distance: dmin=2s+1
Density: r=2s/(22s-1).


[0013] As can be seen from Fig. 17, Euclidean geometry codes have a structure with a cyclically sifted location of "1" in each row from an adjacent row. This structure can characteristically configure codes easily and definitely.

[0014] The check matrix generating method in the second literature further includes changing row and column weights based on the Euclidean geometry codes to extend rows and columns, if necessary. For example, when a column weight in EG (2, 22) is separated into halves, in the second literature, every other one of four weights located in one column is separated into two groups. Fig. 18 is a diagram of an exemplary regular separation of the column weight from 4 into 2.

[0015] On the other hand, M. G. Luby, M. Mitzenmacher, M. A. Shokrollahi, and D. A. Spielman, "Improved Low-Density Parity-Check Codes Using Irregular Graphs and Belief Propagation", Proceedings of 1998 IEEE International Symposium on Information Theory, pp. 171, Cambridge, Mass., August 16-21, 1998" has reported that "irregular-LDPC codes" have a better characteristic than that of "Regular-LDPC Codes". This is theoretically analyzed in T. J. Richardson and R. Urbanke, "The capacity of low-density parity-check codes under message passing decoding", IEEE Trans. Inform. Theory, vol. 47, No.2, pp. 599-618, Feb. 2001" and S.-Y Chung, T. J. Richardson, and R. Urbanke, "Analysis of Sum-Product Decoding of Low-Density Parity-Check Codes Using a Gaussian Approximation", IEEE Trans. Inform. Theory, vol. 47, No.2, pp. 657-670, Feb. 2001". The "irregular-LDPC codes" represent such LDPC codes that have non-uniformity in either or both of row and column weights.

[0016] Particularly, in the fifth literature, a "Sum-Product Algorithm" for LDPC codes is analyzed on the assumption that a log likelihood ratio (LLR) between an input and an output at an iterative decoder can be approximated in a Gaussian distribution, to derive a satisfactory ensemble of row and column weights.

[0017] According to the conventional method of generating check matrixes for LDPC codes disclosed in the fifth literature, however, the number of "1" points in a row (corresponding to a degree distribution of variable nodes described later) and the number of "1" points in a column (corresponding to a degree distribution of check nodes described later) are both employed as variables to derive the degree distribution of variable nodes and the degree distribution of check nodes that can maximize the following equation (1) (rate: coding rate). In other words, a linear programming is employed to search an ensemble that minimizes a Signal to Noise Ratio (SNR).



[0018] Therefore, a problem arises that a check matrix derived from the maximum of the "rate" has a flux and unstable characteristic. In addition, the conventional method of generating check matrixes for LDPC codes iteratively executes the derivation of the degree distribution of variable nodes and the derivation of the degree distribution of check nodes over certain times.
Therefore, a problem arises that it takes time to some extent for searching.

[0019] VASIC B ET AL: "Lattice low-density parity check codes and their application in partial response systems" PROC., IEEE INTERNATIONAL SYMPOSIUM ON INFORMATION THEORY, ISIT 02, LAUSANNE, SWITZERLAND, 30 June 2002, page 453, XP010602164 discloses a combinatorial construction of high rate low-density parity check codes based on two-dimensional integer lattices. A class of codes with a wide range of lengths, column weights and minimum distances is obtained. The resulting codes are doubly circulant, i.e. the matrix of parity checks is an array of circulant matrices. The bounds on the minimum distance are established and the performance of these codes in partial response channels is investigated.

[0020] The present invention has been achieved in consideration of the above problems, and accordingly has an object to provide a method of generating check matrixes for LDPC codes having satisfactory performance capable of easily searching a definite and characteristic stabilized check matrix for LDPC codes corresponding to an optional ensemble, and a check matrix generating apparatus.

[0021] The scope of the invention is defined by the appended claims.

BRIEF DESCRIPTION OF THE DRAWING



[0022] 

Fig. 1 is a flowchart of a method of generating check matrixes for LDPC codes according to the present invention; Fig. 2 is a diagram of an LDPC encoding/decoding system; Fig. 3 is a diagram of a lattice structure when 1 (x, y)=m×x+y+1, m=5, and k=3; Ftg. 4 is a diagram of m classes using a slope s line as a unit; Fig. 5 is a diagram of an algorithm to design an aggregate that sets 8 as a minimum number of cycles; Fig. 6 is a diagram of a search result when the algorithm shown in Fig. 5 is executed using m=5 and k=3; Fig. 7 is a diagram of one example of a basic matrix; Fig. 8 is a diagram of a search result when the algorithm shown in Fig. 5 is executed using m=353 and k=10; Fig. 9 is a diagram of an algorithm of permutation (rearrangement) of a basic matrix; Fig. 10 is a diagram of one example of a basic matrix after a permutation; Fig. 11 is a diagram of a final ensemble of a generator function λ(x) and a generator function ρ(x); Fig. 12 is a diagram of a dividing procedure according to a conventional paper; Fig. 13 is a diagram of a basic random sequence C(i) and permutation patterns of the basic random sequence; Fig. 14 is a diagram of Latin square sequences Ljq (i); Fig. 15 is a diagram of a relation between an Eb/No and a bit error rate; Fig. 16 is a diagram of a conventional check matrix for LDPC codes; Fig. 17 is a diagram of a configuration of Euclidean geometry codes EG (2, 22); and Fig. 18 is a diagram of an exemplary column weight regularly separated from 4 into 2.


BEST MODE FOR CARRYING OUT THE INVENTION



[0023] Exemplary embodiments of a method and an apparatus for generating a check matrix according to the present invention will be explained in detail below with reference to the accompanying drawings. It should be noted that the embodiments are not intended to limit the invention.

[0024] Fig. 1 is a flowchart of a method of generating check matrixes for LDPC codes according to the present invention. The method of generating check matrixes for LDPC codes according to the present embodiment may be executed within a communication apparatus according to set parameters, or may be executed within other control unit (such as a computer) outside of the communication apparatus. When the method of generating check matrixes for LDPC codes according to the present embodiment is executed at the outside of the communication apparatus, generated check matrixes for LDPC codes are stored within the communication apparatus. In the following embodiment, the above method is executed within the communication apparatus for the sake of explanation.

[0025] Prior to explanation of the method of generating check matrixes for LDPC codes according to the present embodiment, the positions of an encoder and a decoder capable of achieving the method are explained first together with the conventional method of generating check matrixes for "irregular-LDPC codes".

[0026] Fig. 2 is a diagram of an LDPC encoding/decoding system. In Fig. 2, a communication apparatus at a sending side includes an encoder 101 and a modulator 102, and a communication apparatus at a receiving side includes a demodulator 104 and a decoder 105. Flows of encoding and decoding using LDPC codes will be explained below.

[0027] At the sending side, the encoder 101 generates a check matrix H using the method of generating check matrixes for LDPC codes according to the present embodiment described later. Then, a generator matrix G is derived from the following condition.





[0028] The encoder 101 then receives a message (m1 m2 ... mk) of an information length k, and generates a code word C using the generator matrix G.



[0029] The modulator 102 subjects the generated code word C to a digital modulation such as BPSK, QPSK, and multi-valued QAM, and sends the modulated signal.

[0030] At the receiving side, on the other hand, the demodulator 104 receives the modulated signal via the channel 103, and subjects it to a digital demodulation such as BPSK, QPSK, and multi-valued QAM. The decoder 105 then subjects the LDPC-coded, demodulated result to an iterative decoding by "sum-product algorithm" and provides an estimated result (corresponding to the original m1 m2 ... mk).

[0031] The conventional method of generating check matrixes for "irregular-LDPC codes" theoretically analyzed in the fifth literature is explained next in detail. In this case, a "sum-product algorithm" for LDPC codes is analyzed, on the assumption that a log likelihood ratio (LLR) between an input and an output at an iterative decoder can be approximated in a Gaussian distribution, to derive a satisfactory ensemble of row and column weights.

[0032] The method of generating check matrixes for LDPC Codes described in the fifth literature, or Gaussian Approximation, has a premise that defines a point of "1" on a row as a variable node and a point of "1" on a column as a check node in the check matrix.

[0033] LLR message propagation from a check node to a variable node is analyzed first. The following function (2) is defined on condition that 0<s<∞ and 0≤t≤∞. In this case, s=mu0 denotes a mean of u0; u0 an LLR associated with a signal received via a channel containing a Gaussian noise of distributed value σn2; and t an ensemble average of LLR output values at check nodes at the time of certain iteration.



[0034] λi and ρi denote ratios of edges belonging to variable nodes and check nodes at a degree of i, respectively. dl denotes a degree of maximum variable nodes, and dr a degree of maximum check nodes. λ(x) and ρ(x) denote generator functions of degree distribution associated with variable nodes and check nodes, and can be represented by the equations (3) and (4), respectively. (A degree is expressed as the number of "1 "s in each row of variable nodes or each column of check nodes).




where, φ(x) is defined as shown in the following equation (5).



[0035] The equation (2) can be represented equivalently by the following equation (6).


where tl denotes an ensemble average of LLR output values on check nodes at the time of the I-th iteration.

[0036] A condition required for deriving an SNR limit (threshold) that provides an error with a value of 0 includes tl(s)→∞ (expressed as R+) when I→∞. In order to satisfy this condition, it is required to satisfy the following conditional equation (7).



[0037] LLR message propagation from a variable node to a check node is analyzed next. The following equation (8) is defined on condition that 0<s<∞ and 0<r≤1. In this case, r has an initial value r0 of φ(s).



[0038] The equation (8) can be represented equivalently by the following equation (9).



[0039] A condition required for deriving an SNR limit (threshold) that provides an error with a value of 0 includes rl(s)→0. In order to satisfy this condition, it is required to satisfy the following conditional equation (10).



[0040] In the fifth literature, optimal degrees are searched for variable nodes and check nodes using the above equation in the following procedure (Gaussian Approximation).
  1. (1) On the assumption that a generator function λ(x) and a Gaussian noise σn are given, a generator function ρ(x) is used as a variable to search a point that maximizes the equation (1) previously described. A constraint condition associated with this search includes normalization to ρ(x)=1 and satisfaction of the equation (7).
  2. (2) On the assumption that a generator function (x) and Gaussian noise σn are given (as a value resulted from the procedure (1), for example), a generator function λ(x) is used as a variable to search a point that maximizes the equation (1). A constraint condition associated with this search includes normalization to λ(x)=1 and satisfaction of the equation (10).
  3. (3) In order to find the maximum "rate", the procedures (1) and (2) are iteratively executed to search a better ensemble of the generator function λ(x) and the generator function ρ(x) with a linear programming.
  4. (4) Finally, signal power is normalized to 1 based on Gaussian noise σn to find an SNR limit (threshold) (see the equation (11)).




[0041] A problem is found in the fifth literature that a check matrix derived from the maximum of the "rate (coding rate)" is flux, and the rate fixed in design as a specification varies. In addition, in the fifth literature, the derivation of the degree distribution associated with variable nodes and the derivation of the degree distribution associated with check nodes are iteratively executes over certain times. Therefore, a problem arises that it takes time to some extent for searching. Further, a problem arises that the check matrix does not easily apply to an optional ensemble, an optional code length, and an optional coding rate.

[0042] In the present embodiment, a method of easily searching in a short time a definite and characteristic-stabilized check matrix for "irregular-LDPC codes", which corresponds to an optional ensemble, an optional code length, and an optional coding rate, is explained (see Fig. 1). Specifically, the check matrix for "irregular-LDPC codes" is generated by using a basic matrix described later (definition: "row and column weights are constant", and "number of cycles is equal to or more than 6").

[0043] In the method of generating a check matrix for LDPC codes according to the present embodiment, a coding rate is determined first (step S1). As an example, the coding rate is set to 0.5.

[0044] A basic matrix based on an integer lattice structure is created on the assumption of the basic matrix (definition: "row and column weights are constant", and "number of cycles is equal to or more than 6") that is necessary to obtain a check matrix of "irregular-LDPC codes" (step S2). In the coding/decoding using LDPC codes, generally when the number of "cycle 4" and "cycle 6" is as small as possible on a binary graph, satisfactory characteristics can be obtained. Therefore, LDPC codes having a construction of restricting the occurrence of a small number of "cycle 4" and "cycle 6" is preferable. Therefore, in the present embodiment, a basic matrix using a minimum number of cycles as 8 is created. A procedure of creating a basic matrix (minimum number of cycles as 8) based on an integer lattice structure will be explained. A basic matrix using a Cayley graph, a basic matrix using a Ramanujan graph, or other matrix can be used so long as the basic matrix satisfies the above definition.

[0045] The procedure of creating a basic matrix based on an integer lattice structure will be explained.
  1. (1) First, an aggregate of lines that connect between points (a combination of points) in an integer lattice structure is designed. For example, an aggregate of a lattice structure is set as L=(x, y). In this case, x is an integer of 0≤x≤k-1, y is an integer of 0≤y≤m-1, k is an integer, and m is a prime number. |(x, y) is a linear mapping to map the aggregate L to an aggregate V of points. Fig. 3 is a diagram of a lattice structure when 1(x, y)=m×x+y+1, m=5, and k=3. In Fig. 3, a combination of points that satisfies a certain condition is called a line (block). For example, a line having a slope s (0≤s≤m-1) consists of a line of l(x, a+sx mod m) having a point (0, a) as a starting point, where a is 0≤a≤m-1. As a result, m classes having the line of the slope s as a unit are created. Fig. 4 is a diagram of m classes having the line of the slope s as a unit.
  2. (2) An aggregate having a minimum number of cycles as 8 is designed based on the algorithm shown in Fig. 5. In other words, a number of columns, a number of rows, weight of columns, and weight of rows are searched. In Fig. 5, S denotes an aggregate of s, and B(s) denotes an aggregate of a class corresponding to the slope s.


[0046] Fig. 6 is a diagram of a search result when the above algorithm is executed using m=5 and k=3. N=|B| denotes a number of columns in the basic matrix, M=|V| denotes a number of rows in the basic matrix, dc denotes weight of rows in the basic matrix, and dv denotes a number of columns in the basic matrix. Fig. 7 is a diagram of an execution result of the above algorithm, that is, a basic matrix.

[0047] The basic matrix needs to be designed in a relatively large size in order to realize a division of rows and columns described later. When the above algorithm is executed using k=10 and m=353, a search result shown in Fig. 8 is obtained. When dv=10 and dc=7 as shown in Fig. 8, a divide processing described later can be carried out. In the present embodiment, the processing at step S2 is executed to make firm a number of columns N (=2471) in the basic matrix based on the integer lattice structure and a number of rows M (=3530) in the basic matrix based on the integer lattice structure.

[0048] Next, a number of columns N' of a check matrix (a check matrix of "irregular-LDPC codes") finally obtained is determined (step S3). At this point, a number of rows M' of the check matrix of "irregular-LDPC codes" is determined as M'=N'× (1-rate). For example, when N'=6000 and rate=0.5, M' is determined as M'=6000×0.5=3000.

[0049] Next, the basic matrix created above is rearranged according to an algorithm shown in Fig. 9 such that 1s are located at higher positions in the columns (step S4). Fig. 9 is a diagram of the algorithm of permutation (rearrangement) of the basic matrix. Fig. 10 is a diagram of one example of a basic matrix after a permutation. The basic matrix when k=3 and m=5 shown in Fig. 7 is rearranged according to the algorithm shown in Fig. 9. Based on this permutation, columns of large weight can be left as far as possible and a variation in the weight within columns can be reduced as far as possible, in delete processing described later.

[0050] Next, an ensemble (degree distribution) of "irregular-LDPC codes" based on a requested coding rate is provisionally obtained using optimization by Gaussian approximation (step S5). In this case, γi(i=1, 2, ..., max, 2≤γ12<...<γmax) denotes a column weight, µ and µ+1 (2≤µ≤dc-1) denote row weights, dc denotes a weight of a basic matrix, λ_γi (0≤λ_γi≤1) denotes a ratio of an edge that belongs to the column weight γi, ρ_µ and ρ_(µ+1) (0≤"ρ_µ, ρ_(µ+1)" ≤1) denote ratios of an edge belonging to the row weights µ and µ+1, b_µ and b_(µ+1) denote nonnegative integers, λ(x) denotes a generator function of a column weight distribution, ρ(x) denotes a generator function of a row weight distribution, n_µ and n_(µ+1) denote numbers of rows of the row weights and µ+1, and n_γi denotes a number of columns of the column weight γi. The above λ(x) and ρ(x) are defined by the equation (13).





[0051] The execution procedure of Gaussian approximation according to the present embodiment to search an ensemble of the generator function λ(x) of a column weight distribution and the generator function ρ(x) of a row weight distribution will be explained.
  1. (1) A coding rate "rate" is fixed (step S1).
  2. (2) A generator function λ(x) and a generator function ρ(x) are simultaneously used as variables, and a linear programming is employed to search optimal generator functions λ(x) and ρ(x) that can maximize Gaussian noise σn (see the following equation (14)). A constraint condition associated with this search is to satisfy the equation (18) described later.



[0052] The above s denotes an average of a log likelihood ratio (LLR) between a binary signal of {-1, 1} received as a transmission signal and a signal received through the Gaussian channel, and can be derived from S=2/σn2.

[0053] As explained above, according to the present embodiment, the generator functions λ(x) and ρ(x) that satisfy a predetermined condition are obtained at one-time linear programming. Therefore, it is possible to create a definite and characteristic-stabilized ensemble more easily in a shorter time than it is by the method described in the fifth literature that iteratively executes derivation of the generator functions λ(x) and ρ(x) to derive both optimal values. The number of rows M' (=3000) is obtained after executing the divide processing of rows described later following b_µ, b_(µ+1), µ, and µ+1 obtained at step S5. The number of rows M (=3530) of the basic matrix is obtained. In this case, rows of a number shown in the following equation (15) are deleted in order from the bottom of the basic matrix after the permutation (step S6). In this example, (3530-3000)/(1+0)=530 rows are deleted. A matrix after deleting the rows has a set of row weights as {d1, d2, ..., dv}.



[0054] Next, an ensemble (degree distribution) of "irregular-LDPC codes" based on a requested coding rate is obtained using optimization by Gaussian approximation, using the following constraint equations (16), (17), (18), and (19) (step S7). A matrix β={2, 3, ..., dv} expressed by (β)i,j of the equation (16) represents a matrix of a nonnegative integer |(β)×L that includes all elements satisfying the equation (16). A matrix shown expressed by (β)i,j of the equation (17) represents a square matrix of a nonnegative integer |×|.









[0055] Fig. 11 is a diagram of a final ensemble of the generator functions λ(x) and ρ(x) obtained at step S7 when the ensemble is adjusted in the above procedure.

[0056] Finally, a dividing procedure of one row or one column in the basic matrix after the permutation (step S8) will be explained. The second literature proposes a regular dividing method concerning the dividing procedure. Fig. 12 is a diagram of the dividing procedure in the literature. First, a matrix is numbered as shown in Fig. 12. In this example, column numbers are given as 1, 2, 3, and so on in order from the left end, and row numbers are given as 1, 2, 3, and so on in order from the top. For dividing 32 points×one column into 8 points×4 columns, for example, this is regularly divided according to the following equation (20).



[0057] In the above equation, m=1, 2, 3, and 4, and n=0, 1, 2, 3, 4, 5, 6, and 7, and I denotes a column number of EG (2,25). Bl(x) denotes a position of "1" in an I-th column of EG (2,25). Sm(n) denotes a position of "1" in an m-th column of the matrix after the division.

[0058] Specifically, a row number that shows a position of "1" in the I-th column of EG (2,25) is as follows. B1(x)={1 32 114 136 149 223 260 382 402 438 467 507 574 579 588 622 634 637 638 676 717 728 790 851 861 879 947 954 971 977 979 998}. As a result, a row number that shows a position of "1" in the first to fourth columns of the matrix after the division is as follows, based on a regular extraction of the number of "1" from Bi(x).








In other words, 32 points×one column is divided into 8 points×4 columns.

[0059] On the other hand, in the divide processing of the basic matrix after the permutation according to the present invention, a regular dividing like the above processing is not executed but the number of "1" is extracted at random from Bi(x) (see a detailed example of a random division described later). Any method of extraction processing can be used so long as randomness is maintained.

[0060] When a position of "1" in the m-th column of the matrix after the division is Rm(n), for example, Rm(n) becomes as follows.









[0061] One example of a random division, that is, "a division method using a Latin square of random sequences", will be explained in detail. A random sequence to carry out the random division is created easily and definitely. An advantage of this method is that the same random sequence can be created at the sending side and the receiving side. This is extremely important in a real system. This method also has an advantage that a condition of code characteristic can be accurately prescribed.
  1. (1) Creation of basic random sequences:
    An example of random sequence creation is described below using Euclidean geometry codes EG (2,25) for convenience of explanation. In Euclidean geometry codes EG (2,25), the number of "1"s present in a row is equal to 25=32.
    When P is used for the minimum prime number that satisfies P≥dv=2s, for example, P=37 in the case of dv=25. A basic random sequence C(i) with a sequence length, P-5=32, is created in accordance with the equation (21). dv denotes a maximum weight of a column. Therefore, when codes other than Euclidean geometry codes are selected as a basic matrix, the use of dv of this basic matrix makes it possible to apply this division.


    where, i=0, 1, ..., P-2; and Go denotes an original source of Galois Field GF(P). As a result, C(i) is represented by the following equation:

  2. (2) Numbers larger than 32 are deleted so as to obtain the sequence length dv=25=32.

  3. (3) A permutation pattern LBj(i) is created using the following equation (22).


    A number larger than LBj(i) is deleted. Fig. 13 is a diagram of the basic random sequence C(i) and the permutation pattern LBj(i) of the basic random sequence.
  4. (4) A j-th Latin square matrix Ljp (i) in the column q and the row i is calculated from the following equation (23) to execute a division. When the column weight d_β is d_β<dv, numbers larger than d_β are thinned from the elements of Ljq(i), based on the delete processing at step S6.





[0062] Fig. 14 is a diagram of Latin square sequences Ljq (i). The Latin square sequences Ljq (i) are used to determine a division pattern of the j×32+q-th column of the matrix to be expanded. For example, the 670-th column g670 (1) of EG (2,25) to be shortened by deletion is determined as follows:


This is divided into five columns of the weight 6 and one column of the weight 2.
The corresponding Latin square Ljq (i) is 20x32+30=670. Therefore, the following Latin square is obtained:


As a result, the division pattern becomes as follows:












In general, the elements Lj,q (i) of Latin square for gc,e(I) are determined based on the following equation (24).



[0063] Characteristics of the above LDPC codes will be compared below. Fig. 15 is a diagram of a relation between an Eb/No (a signal power to noise power ratio per one information bit) and a bit error rate (BER). A decoding method is "Sum-Product algorithm". This characteristic uses the ensemble shown in Fig. 11. Fig. 15 is a characteristic comparison between the execution of the regular division as described in the second literature and the execution of the division according to a Latin square of random sequences.

[0064] As is clear from Fig. 15, according to the regular division as described in the second literature, a large improvement cannot be expected even with "irregular-LDPC codes". In contrast, the random division of the present embodiment can provide a remarkably improved performance when it is implemented because the probability of the occurrence of a loop decreases substantially.

[0065] As explained above, according to the present embodiment, first, a coding rate is determined. Next, a "basic matrix based on an integer lattice structure" having constant weights of rows and columns and a minimum number of cycles as 8 is created. The created basic matrix is substituted based on a specific relational equation. An ensemble of irregular-LDPC codes is provisionally searched by Gaussian approximation based on a condition before row deletion. Rows are deleted in order from the bottom of the basic matrix after the permutation by considering the number of rows after a division. An optimal ensemble of irregular-LDPC codes is searched by Gaussian approximation based on a condition after the row deletion. Finally, the weight of the basic matrix after the row deletion is divided at random according to a predetermined procedure based on this ensemble. With this arrangement, a definite and characteristic-stabilized check matrix for "irregular-LDPC codes" can be generated easily in a short time corresponding to an optional ensemble, an optional code length, and an optional coding rate.

[0066] As explained above, according to the present invention, first, a coding rate is determined. Next, a basic matrix having constant weights of rows and columns and a number of cycles equal to or larger than 6 is created. The created basic matrix is substituted based on a specific relational equation. An ensemble of irregular-LDPC codes is provisionally searched by Gaussian approximation based on a condition before row deletion. Rows are deleted in order from the bottom of the basic matrix after the permutation by considering the number of rows after a division. An optimal ensemble of irregular-LDPC codes is searched by Gaussian approximation based on a condition after the row deletion. Finally, the weight of the basic matrix after the row deletion is divided at random according to a predetermined procedure based on this optimal ensemble. With this arrangement, there is an effect that a definite and characteristic-stabilized check matrix for "irregular-LDPC codes" can be generated easily in a short time corresponding to an optional ensemble, an optional code length, and an optional coding rate.

INDUSTRIAL APPLICABILITY



[0067] As explained above, the method of generating check matrixes for LDPC codes and the check matrix generating apparatus according to the present invention are useful for the communication system that employs the LDPC codes as error correcting codes. Particularly, the method and the apparatus are suitable for a communication apparatus that searches definite and characteristic-stabilized LDPC codes.


Claims

1. A communication method comprising the method steps of:

generating a check matrix for an irregular low-density parity-check code;

using the generated check matrix for encoding a message thereby generating a code word; and

modulating the generated code word;

wherein the method step of generating a check matrix comprises the following step:

determining (S1) a coding rate; and is characterised by further comprising:

generating (S2) a basic matrix that satisfies conditions that "weights of rows and columns are constant" and "the length of the smallest cycle is equal to or more than six";

determining (S3) number of columns and number of rows of the check matrix to be finally obtained;

rearranging (S4) rows of the generated basic matrix, based on a specific relational equation;

obtaining (S5) provisionally an ensemble of row weights and column weights of the low-density parity check code by executing an optimization by Gaussian approximation based on a predetermined condition before a row deletion;

deleting (S6) rows of the rearranged basic matrix in order from the bottom by considering the number of rows after a division, that is, the number of rows of the check matrix to be finally obtained;

obtaining (S7) an optimal ensemble of row weights and column weights of the low-density parity check code by executing the optimization by Gaussian approximation based on a predetermined condition after the row deletion; and

dividing (S8) at random the rows and columns of the basic matrix after the row deletion based on the optimal ensemble, wherein:

a basic matrix is generated based on an integer lattice structure that satisfies conditions that "weights of rows and columns are constant" and "the length of the smallest cycle is eight", as the basic matrix that satisfies the conditions that "weights of rows and columns are constant" and "the length of the smallest cycle is equal to or more than six".


 
2. The method according to claim 1, wherein the specific relational equation used at the method step of rearranging is an equation that can rearrange the rows of the generated basic matrix such that 1s are located at higher positions in the columns.
 
3. The method according to claim 1, wherein, in the Gaussian approximation, an ensemble of optimum row weights and an ensemble of optimum column weights are obtained using a single linear programming to maximize a Gaussian noise with fixed coding rate.
 
4. The method according to any one of claims 1 to 3, wherein, at the method step of dividing, a Latin square of a basic random sequence is generated, and a "1" is extracted from each row and each column in the basic matrix after the row deletion, thereby dividing each column and each row at random based on the Latin square.
 
5. An apparatus comprising:

means for generating a check matrix for an irregular low-density parity-check code;

an encoder adapted to use the generated check matrix for encoding a message thereby adapted to generate a code word; and

a modulator for modulating the generated code word;

the means for generating a check matrix for an irregular low-density parity-check code comprising:

a coding-rate determining unit configured to determine a coding rate;

characterised by further comprising:

a basic-matrix generating unit configured to generate a basic matrix that satisfies conditions that "weights of rows and columns are constant" and "the length of the smallest cycle is equal to or more than six";

a rearranging unit configured to rearrange rows of the generated basic matrix, based on a specific relational equation;

a first weight obtaining unit configured to obtain provisionally an ensemble of row weights and column weights of the low-density parity check code by executing an optimization by Gaussian approximation based on a predetermined condition before a row deletion;

a row deleting unit configured to delete rows of the rearranged basic matrix in order from the bottom by considering the number of rows after a division, that is, the number of rows of the check matrix to be finally obtained;

a second weight obtaining unit configured to obtain an optimal ensemble of row weights and column weights of the low-density parity check code by executing the optimization by Gaussian approximation based on a predetermined condition after the row deletion; and

a dividing unit configured to divide at random the rows and columns of the basic matrix after the row deletion based on the optimal ensemble; wherein:

said basic-matrix generating unit is configured to generate a basic matrix based on an integer lattice structure that satisfies conditions that "weights of rows and columns are constant" and "the length of the smallest cycle is eight" as the basic matrix that satisfies the conditions that "weights of rows and columns are constant" and "the length of the smallest cycle is equal to or more than six".


 


Ansprüche

1. Kommunikationsverfahren, das die folgenden Verfahrensschritte umfasst:

Erzeugen einer Prüfmatrix für einen irregulären Paritätsprüfungscode mit niedriger Dichte;

Verwenden der erzeugten Prüfmatrix zum Codieren einer Nachricht und dadurch zum Erzeugen eines Codeworts; und

Modulieren des erzeugten Codeworts;

wobei der Verfahrensschritt des Erzeugens einer Prüfmatrix den folgenden Schritt umfasst:

Bestimmen (S1) einer Codierungsrate; dadurch gekennzeichnet, dass es ferner umfasst:

Erzeugen (S2) einer Grundmatrix, die den Bedingungen genügt, dass "Gewichte von Zeilen und Spalten konstant sind" und dass "die Länge des kleinsten Zyklus gleich oder größer sechs ist";

Bestimmen (S3) der Anzahl von Spalten und der Anzahl von Zeilen der schließlich zu erhaltenden Prüfmatrix;

Umordnen (S4) der Zeilen der erzeugten Grundmatrix auf der Grundlage einer spezifischen Relationsgleichung;

vorläufiges Erhalten (S5) eines Ensembles von Zeilengewichten und Spaltengewichten des Paritätsprüfungscodes mit niedriger Dichte durch Ausführen einer Optimierung durch Gauß'sche Näherung auf der Grundlage einer vorgegebenen Bedingung vor einer Zeilenlöschung;

Löschen (S6) von Zeilen der umgeordneten Grundmatrix der Reihe nach von unten durch Betrachtung der Anzahl der Zeilen nach einer Teilung, d. h. der Anzahl der Zeilen der schließlich zu erhaltenden Prüfmatrix;

Erhalten (S7) eines optimalen Ensembles von Zeilengewichten und Spaltengewichten des Paritätsprüfungscodes mit niedriger Dichte durch Ausführen der Optimierung durch Gauß'sche Näherung auf der Grundlage einer vorgegebenen Bedingung nach der Zeilenlöschung; und

zufälliges Teilen (S8) der Zeilen und Spalten der Grundmatrix nach der Zeilenlöschung auf der Grundlage des optimalen Ensembles, wobei:

als die Grundmatrix, die den Bedingungen genügt, dass "Gewichte von Zeilen und Spalten konstant sind" und dass "die Länge des kleinsten Zyklus gleich oder größer sechs ist", eine Grundmatrix auf der Grundlage einer ganzzahligen Gitterstruktur erzeugt wird, die den Bedingungen genügt, dass "Gewichte von Zeilen und Spalten konstant sind" und dass "die Länge des kleinsten Zyklus acht ist".


 
2. Verfahren nach Anspruch 1, bei dem die in dem Verfahrensschritt des Umordnens verwendete spezifische Relationsgleichung eine Gleichung ist, die die Zeilen der erzeugten Grundmatrix in der Weise umordnen kann, dass sich an höheren Stellen in den Spalten 1 en befinden.
 
3. Verfahren nach Anspruch 1, bei dem in der Gauß'schen Näherung ein Ensemble optimaler Zeilengewichte und ein Ensemble optimaler Spaltengewichte unter Verwendung einer einzelnen linearen Programmierung erhalten werden, um ein Gauß'sches Rauschen mit fester Codierungsrate zu maximieren.
 
4. Verfahren nach einem der Ansprüche 1 bis 3, bei dem in dem Verfahrensschritt des Teilens ein lateinisches Quadrat einer Grundzufallsfolge erzeugt wird und aus jeder Zeile und jeder Spalte in der Grundmatrix nach der Zeilenlöschung eine "1" extrahiert wird, um dadurch jede Spalte und jede Zeile auf der Grundlage des lateinischen Quadrats zufällig zu teilen.
 
5. Vorrichtung, die umfasst:

ein Mittel zum Erzeugen einer Prüfmatrix für einen irregulären Paritätsprüfungscode mit niedriger Dichte;

einen Codierer, der zum Verwenden der erzeugten Prüfmatrix zum Codieren einer Nachricht ausgelegt ist und dadurch zum Erzeugen eines Codeworts ausgelegt ist; und

einen Modulator zum Modulieren des erzeugten Codeworts;

das Mittel zum Erzeugen einer Prüfmatrix für einen irregulären Paritätsprüfungscode mit niedriger Dichte, wobei das Mittel umfasst:

eine Codierungsratenbestimmungseinheit, die zum Bestimmen einer Codierungsrate konfiguriert ist;

dadurch gekennzeichnet, dass sie ferner umfasst:

eine Grundmatrixerzeugungseinheit, die zum Erzeugen einer Grundmatrix konfiguriert ist, die den Bedingungen genügt, dass "Gewichte von Zeilen und Spalten konstant sind" und dass "die Länge des kleinsten Zyklus gleich oder größer sechs ist";

eine Umordnungseinheit, die zum Umordnen der Zeilen der erzeugten Grundmatrix auf der Grundlage einer spezifischen Relationsgleichung konfiguriert ist;

eine erste Einheit zum Erhalten von Gewichten, die zum vorläufiges Erhalten eines Ensembles von Zeilengewichten und Spaltengewichten des Paritätsprüfungscodes mit niedriger Dichte durch Ausführen einer Optimierung durch Gauß'sche Näherung auf der Grundlage einer vorgegebenen Bedingung vor einer Zeilenlöschung konfiguriert ist;

eine Zeilenlöscheinheit, die zum Löschen von Zeilen der umgeordneten Grundmatrix der Reihe nach von unten durch Betrachtung der Anzahl der Zeilen nach einer Teilung, d. h. der Anzahl der Zeilen der schließlich zu erhaltenden Prüfmatrix, konfiguriert ist;

eine zweite Einheit zum Erhalten von Gewichten, die zum Erhalten eines optimalen Ensembles von Zeilengewichten und Spaltengewichten des Paritätsprüfungscodes mit niedriger Dichte durch Ausführen der Optimierung durch Gauß'sche Näherung auf der Grundlage einer vorgegebenen Bedingung nach der Zeilenlöschung konfiguriert ist; und

eine Teilungseinheit, die zum zufälligen Teilen der Zeilen und Spalten der Grundmatrix nach der Zeilenlöschung auf der Grundlage des optimalen Ensembles konfiguriert ist; wobei:

die Grundmatrixerzeugungseinheit zum Erzeugen einer Grundmatrix auf der Grundlage einer ganzzahligen Gitterstruktur konfiguriert ist, die als die Grundmatrix, die den Bedingungen genügt, dass "Gewichte von Zeilen und Spalten konstant sind" und dass "die Länge des kleinsten Zyklus gleich oder größer sechs ist", den Bedingungen genügt, dass "Gewichte von Zeilen und Spalten konstant sind" und dass "die Länge des kleinsten Zyklus acht ist".


 


Revendications

1. Procédé de communication comprenant les étapes consistant à :

générer une matrice de contrôle pour un code de contrôle de parité à faible densité irrégulier ;

utiliser la matrice de contrôle générée de façon à coder un message en générant de ce fait un mot de code ; et

moduler le mot de code généré ;

dans lequel l'étape consistant à générer une matrice de contrôle comprend l'étape suivante consistant à :

déterminer (S1) une vitesse de codage ; et il est caractérisé par le fait qu'il comprend en outre les étapes consistant à :

générer (S2) une matrice de base qui satisfait aux conditions selon lesquelles « les poids des lignes et des colonnes sont constants » et « la longueur du plus petit cycle est égale ou supérieure à six » ;

déterminer (S3) le nombre de colonnes et le nombre de lignes de la matrice de contrôle à obtenir à la fin ;

réarranger (S4) les lignes de la matrice de base générée, sur la base d'une équation relationnelle spécifique ;

obtenir (S5) provisoirement un ensemble de poids de ligne et de poids de colonne du code de contrôle de parité à faible densité en exécutant une optimisation par une approximation gaussienne sur la base d'un état prédéterminé avant une suppression de ligne ;

supprimer (S6) les lignes de la matrice de base réarrangée dans l'ordre à partir du bas en prenant en considération le nombre de lignes après une division, à savoir, le nombre de lignes de la matrice de contrôle à obtenir à la fin ;

obtenir (S7) un ensemble optimal de poids de ligne et de poids de colonne du code de contrôle de parité à faible densité en exécutant l'optimisation par une approximation gaussienne sur la base d'un état prédéterminé après la suppression de ligne ; et

diviser (S8) au hasard les lignes et les colonnes de la matrice de base après la suppression de ligne sur la base de l'ensemble optimal, dans lequel :

une matrice de base est générée sur la base d'une structure en treillis de nombres entiers qui satisfait aux conditions selon lesquelles « les poids des lignes et des colonnes sont constants » et « la longueur du plus petit cycle est égale à huit », comme matrice de base qui satisfait aux conditions selon lesquelles « les poids des lignes et des colonnes sont constants » et « la longueur du plus petit cycle est égale ou supérieure à six ».


 
2. Procédé selon la revendication 1, dans lequel l'équation relationnelle spécifique utilisée dans l'étape du réarrangement est une équation qui permet de réarranger les lignes de la matrice de base générée de telle sorte que des 1 se situent aux positions hautes dans les colonnes.
 
3. Procédé selon la revendication 1, dans lequel, dans l'approximation gaussienne, un ensemble de poids de ligne optimaux et un ensemble de poids de colonne optimaux sont obtenus en utilisant une seule programmation linéaire de façon à maximiser un bruit gaussien avec une vitesse de codage fixe.
 
4. Procédé selon l'une quelconque des revendications 1 à 3, dans lequel, dans l'étape de division, un carré latin d'une séquence aléatoire de base est généré, et un « 1 » est extrait à partir de chaque ligne et de chaque colonne dans la matrice de base après la suppression de ligne, en divisant de ce fait chaque colonne et chaque ligne au hasard sur la base du carré latin.
 
5. Appareil comprenant :

des moyens destinés à générer une matrice de contrôle pour un code de contrôle de parité à faible densité irrégulier ;

un codeur adapté pour utiliser la matrice de contrôle générée de façon à coder un message adapté de ce fait pour générer un mot de code ; et

un modulateur destiné à moduler le mot de code généré ;

les moyens destinés à générer une matrice de contrôle pour un code de contrôle de parité à faible densité irrégulier comprenant ;

une unité de détermination de vitesse de codage configurée de façon à déterminer une vitesse de codage ;

caractérisé par le fait qu'il comprend en outre :

une unité de génération de matrice de base configurée de façon à générer une matrice de base qui satisfait aux conditions selon lesquelles « les poids des lignes et des colonnes sont constants » et « la longueur du plus petit cycle est égale ou supérieure à six » ;

une unité de réarrangement destinée à réarranger les lignes de la matrice de base générée, sur la base d'une équation relationnelle spécifique ;

une première unité d'obtention de poids configurée de façon à obtenir provisoirement un ensemble de poids de ligne et de poids de colonne du code de contrôle de parité à faible densité en exécutant une optimisation par une approximation gaussienne sur la base d'un état prédéterminé avant une suppression de ligne ;

une unité de suppression de ligne configurée de façon à supprimer les lignes de la matrice de base réarrangée dans l'ordre à partir du bas en prenant en considération le nombre de lignes après une division, à savoir, le nombre de lignes de la matrice de contrôle à obtenir à la fin ;

une seconde unité d'obtention de poids configurée de façon à obtenir un ensemble optimal de poids de ligne et de poids de colonne du code de contrôle de parité à faible densité en exécutant l'optimisation par une approximation gaussienne sur la base d'un état prédéterminé après la suppression de ligne ; et

une unité de division configurée de façon à diviser au hasard les lignes et les colonnes de la matrice de base après la suppression de ligne sur la base de l'ensemble optimal ; dans lequel :

ladite unité de génération de matrice de base est configurée de façon à générer une matrice de base sur la base d'une structure en treillis de nombres entiers qui satisfait aux conditions selon lesquelles « les poids des lignes et des colonnes sont constants » et « la longueur du plus petit cycle est égale à huit », comme matrice de base qui satisfait aux conditions selon lesquelles « les poids des lignes et des colonnes sont constants » et « la longueur du plus petit cycle est égale ou supérieure à six ».


 




Drawing



































Cited references

REFERENCES CITED IN THE DESCRIPTION



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Non-patent literature cited in the description