Eliminating trapping sets in low-density parity-check codes by using Tanner graph covers

We discuss error floor asympotics and present a method for improving the performance of low-density parity-check (LDPC) codes in the high SNR (error floor) region. The method is based on Tanner graph covers that do not have trapping sets from the original code. The advantages of the method are that it is universal, as it can be applied to any LDPC code/channel/decoding algorithm and it improves performance at the expense of increasing the code length, without losing the code regularity, without changing the decoding algorithm, and, under certain conditions, without lowering the code rate. The proposed method can be modified to construct convolutional LDPC codes also. The method is illustrated by modifying Tanner, MacKay and Margulis codes to improve performance on the binary symmetric channel (BSC) under the Gallager B decoding algorithm. Decoding results on AWGN channel are also presented to illustrate that optimizing codes for one channel/decoding algorithm can lead to performance improvement on other channels.

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

[2]  R. M. Tanner,et al.  A Class of Group-Structured LDPC Codes , 2001 .

[3]  P. Vontobel,et al.  Graph-Cover Decoding and Finite-Length Analysis of Message-Passing Iterative Decoding of LDPC Codes , 2005, ArXiv.

[4]  Ali Emre Pusane,et al.  Pseudocodeword Performance Analysis for LDPC Convolutional Codes , 2006, IEEE Transactions on Information Theory.

[5]  R. Koetter,et al.  On the Effective Weights of Pseudocodewords for Codes Defined on Graphs with Cycles , 2001 .

[6]  Lara Dolecek,et al.  GEN03-6: Investigation of Error Floors of Structured Low-Density Parity-Check Codes by Hardware Emulation , 2006, IEEE Globecom 2006.

[7]  N. Varnica,et al.  Improvements in belief-propagation decoding based on averaging information from decoder and correction of clusters of nodes , 2006, IEEE Communications Letters.

[8]  David J. C. MacKay,et al.  Encyclopedia of Sparse Graph Codes , 1999 .

[9]  Shashi Kiran Chilappagari,et al.  Error Floors of LDPC Codes on the Binary Symmetric Channel , 2006, 2006 IEEE International Conference on Communications.

[10]  Roxana Smarandache,et al.  Pseudo-Codeword Analysis of Tanner Graphs From Projective and Euclidean Planes , 2006, IEEE Transactions on Information Theory.

[11]  Ali Emre Pusane,et al.  On Deriving Good LDPC Convolutional Codes from QC LDPC Block Codes , 2007, 2007 IEEE International Symposium on Information Theory.

[12]  Shen Qiong-xia.  Design of Structured LDPC Codes with Large Girth , 2009 .

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

[14]  J. Thorpe.  Low-Density Parity-Check (LDPC) Codes Constructed from Protographs , 2003 .

[15]  H. Vincent Poor,et al.  Upper Bounding the Performance of Arbitrary Finite LDPC Codes on Binary Erasure Channels , 2006, 2006 IEEE International Symposium on Information Theory.

[16]  Johannes B. Huber,et al.  CTH02-4: When Does One Redundant Parity-Check Equation Matter? , 2006, IEEE Globecom 2006.

[17]  Chih-Chun Wang.  Code Annealing and the Suppressing Effect of the Cyclically Lifted LDPC Code Ensemble , 2006, 2006 IEEE Information Theory Workshop - ITW '06 Chengdu.

[18]  Michael Chertkov,et al.  Instanton analysis of Low-Density Parity-Check codes in the error-floor regime , 2006, 2006 IEEE International Symposium on Information Theory.

[19]  Niclas Wiberg,et al.  Codes and Decoding on General Graphs , 1996 .

[20]  Joachim Rosenthal,et al.  Tree-Based Construction of LDPC Codes Having Good Pseudocodeword Weights , 2005, IEEE Transactions on Information Theory.

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

[22]  J. Moura,et al.  Structured low-density parity-check codes , 2004, IEEE Signal Processing Magazine.

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

[24]  Emre Telatar,et al.  Finite-length analysis of low-density parity-check codes on the binary erasure channel , 2002, IEEE Trans. Inf. Theory.

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

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