On the computation of the minimum distance of low-density parity-check codes

Low-density parity-check (LDPC) codes in their broader-sense definition are linear codes whose parity-check matrices have fewer 1s than 0s. Finding their minimum distance is therefore in general an NP-hard problem. We propose a randomized algorithm called nearest nonzero codeword search (NNCS) approach to tackle this problem for iteratively decodable LDPC codes. The principle of the NNCS approach is to search codewords locally around the all-zero codeword perturbed by minimal noise, anticipating that the resultant nearest nonzero codewords will most likely contain the minimum-Hamming- weight codeword whose Hamming weight is equal to the minimum distance of the linear code. This approach has its roots in Berrou et al.'s error-impulse method and a form of Fossorier's list decoding for LDPC codes.

[1]  D.J.C. MacKay,et al.  Good error-correcting codes based on very sparse matrices , 1997, Proceedings of IEEE International Symposium on Information Theory.

[2]  Evangelos Eleftheriou,et al.  Regular and irregular progressive edge-growth tanner graphs , 2005, IEEE Transactions on Information Theory.

[3]  Claude Berrou,et al.  Computing the minimum distance of linear codes by the error impulse method , 2002, Global Telecommunications Conference, 2002. GLOBECOM '02. IEEE.

[4]  Hassan Masum.  Review of Algorithmics for hard problems: introduction to combinatorial optimization, randomization, approximation, and heuristics by Juraj Hromkovič. Springer 2001 , 2003, SIGA.

[5]  Rüdiger L. Urbanke,et al.  Design of capacity-approaching irregular low-density parity-check codes , 2001, IEEE Trans. Inf. Theory.

[6]  Thomas J. Richardson,et al.  Error Floors of LDPC Codes , 2003 .

[7]  Pascal O. Vontobel,et al.  Algebraic coding for iterative decoding , 2003 .

[8]  P. Vontobel,et al.  Graph-covers and iterative decoding of nite length codes , 2003 .

[9]  Alexander Vardy,et al.  The intractability of computing the minimum distance of a code , 1997, IEEE Trans. Inf. Theory.

[10]  Evangelos Eleftheriou,et al.  Progressive edge-growth Tanner graphs , 2001, GLOBECOM'01. IEEE Global Telecommunications Conference (Cat. No.01CH37270).

[11]  Sarah J. Johnson,et al.  Regular low-density parity-check codes from combinatorial designs , 2001, Proceedings 2001 IEEE Information Theory Workshop (Cat. No.01EX494).

[12]  Marc P. C. Fossorier,et al.  Iterative reliability-based decoding of low-density parity check codes , 2001, IEEE J. Sel. Areas Commun..

[13]  Robert Michael Tanner,et al.  A recursive approach to low complexity codes , 1981, IEEE Trans. Inf. Theory.

[14]  Amir H. Banihashemi,et al.  A heuristic search for good low-density parity-check codes at short block lengths , 2001, ICC 2001. IEEE International Conference on Communications. Conference Record (Cat. No.01CH37240).

[15]  David J. C. MacKay,et al.  Weaknesses of Margulis and Ramanujan-Margulis low-density parity-check cCodes , 2003, MFCSIT.

[16]  M. Sudan,et al.  Hardness of approximating the minimum distance of a linear code , 2000, 2000 IEEE International Symposium on Information Theory (Cat. No.00CH37060).

[17]  Robert G. Gallager.  Low Density Parity Check Code , 1962 .

[18]  T. Richardson,et al.  Design of provably good low-density parity check codes , 2000, 2000 IEEE International Symposium on Information Theory (Cat. No.00CH37060).

[19]  Robert Michael Tanner,et al.  Minimum-distance bounds by graph analysis , 2001, IEEE Trans. Inf. Theory.

[20]  Shu Lin,et al.  Low-density parity-check codes based on finite geometries: A rediscovery and new results , 2001, IEEE Trans. Inf. Theory.

[21]  Bane V. Vasic,et al.  Structured iteratively decodable codes based on Steiner systems and their application in magnetic recording , 2001, GLOBECOM'01. IEEE Global Telecommunications Conference (Cat. No.01CH37270).

[22]  Jakov Snyders,et al.  Reliability-Based Syndrome Decoding of Linear Block Codes , 1998, IEEE Trans. Inf. Theory.

[23]  G.D. Forney,et al.  Codes on graphs: Normal realizations , 2000, IEEE Trans. Inf. Theory.

[24]  Xiao-Yu Hu,et al.  Irregular progressive edge-growth (PEG) Tanner graphs , 2002, Proceedings IEEE International Symposium on Information Theory,.

[25]  Juraj Hromkovic,et al.  Algorithmics for hard problems - introduction to combinatorial optimization, randomization, approximation, and heuristics , 2001 .

[26]  Robert G. Gallager,et al.  Low-density parity-check codes , 1962, IRE Trans. Inf. Theory.

[27]  Elwyn R. Berlekamp,et al.  On the inherent intractability of certain coding problems (Corresp.) , 1978, IEEE Trans. Inf. Theory.

[28]  David J. C. MacKay,et al.  Good Codes Based on Very Sparse Matrices , 1995, IMACC.

[29]  Xiao-Yu Hu.  Low-delay low-complexity error-correcting codes on sparse graphs , 2003 .

[30]  Pascal O. Vontobel,et al.  Construction of codes based on finite generalized quadrangles for iterative decoding , 2001, Proceedings. 2001 IEEE International Symposium on Information Theory (IEEE Cat. No.01CH37252).

[31]  C. Berrou,et al.  Computing the minimum distances of linear codes by the error impulse method , 2002, Proceedings IEEE International Symposium on Information Theory,.

[32]  A. Glavieux,et al.  Near Shannon limit error-correcting coding and decoding: Turbo-codes. 1 , 1993, Proceedings of ICC '93 - IEEE International Conference on Communications.

[33]  Brendan J. Frey,et al.  Factor graphs and the sum-product algorithm , 2001, IEEE Trans. Inf. Theory.

[34]  Rüdiger L. Urbanke,et al.  The capacity of low-density parity-check codes under message-passing decoding , 2001, IEEE Trans. Inf. Theory.

[35]  P. Vontobel,et al.  Constructions of LDPC Codes using Ramanujan Graphs and Ideas from Margulis , 2000 .