Gallager-type bounds for non-binary linear block codes over memoryless symmetric channels

The performance analysis of non-binary linear block codes is studied under ML decoding where it is assumed that the transmission takes place over memoryless symmetric channels. Gallager-type bounds are derived, and the proposed bounds are exemplified for expurgated regular ensembles of non-binary low-density parity-check (LDPC) codes. These bounds are also compared with classical and recent improved sphere-packing bounds, indicating that these bounding techniques are informative for the performance evaluation of coded communication systems which incorporate non-binary coding techniques.

[1]  David Burshtein,et al.  On the application of LDPC codes to arbitrary discrete-memoryless channels , 2003, IEEE Transactions on Information Theory.

[2]  Shlomo Shamai,et al.  Improved upper bounds on the ensemble performance of ML decoded low density parity check codes , 2000, IEEE Communications Letters.

[3]  Cong Ling,et al.  New Gallager Bounds in Block-Fading Channels , 2007, IEEE Transactions on Information Theory.

[4]  Sheng Tong,et al.  Tangential Sphere Bounds on the Ensemble Performance of ML Decoded Gallager Codes via Their Exact Ensemble Distance Spectrum , 2008, 2008 IEEE International Conference on Communications.

[5]  Shlomo Shamai,et al.  Tightened Upper Bounds on the ML Decoding Error Probability of Binary Linear Block Codes , 2006, 2006 IEEE International Symposium on Information Theory.

[6]  H. Vincent Poor,et al.  Finite-Dimensional Bounds on ${\BBZ}_m$ and Binary LDPC Codes With Belief Propagation Decoders , 2006, IEEE Transactions on Information Theory.

[7]  Masoud Salehi,et al.  New performance bounds for turbo codes , 1998, IEEE Trans. Commun..

[8]  Masoud Salehi,et al.  Performance bounds for turbo-coded modulation systems , 1999, IEEE Trans. Commun..

[9]  Andrew J. Viterbi,et al.  Principles of Digital Communication and Coding , 1979 .

[10]  Li Zhang,et al.  Non-binary LDPC codes vs. Reed-Solomon codes , 2008, 2008 Information Theory and Applications Workshop.

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

[12]  Shlomo Shamai,et al.  Performance Analysis of Linear Codes under Maximum-Likelihood Decoding: A Tutorial , 2006, Found. Trends Commun. Inf. Theory.

[13]  D. Divsalar.  A Simple Tight Bound on Error Probability of Block Codes with Application to Turbo Codes , 1999 .

[14]  Shlomo Shamai,et al.  Tight exponential upper bounds on the ML decoding error probability of block codes over fully interleaved fading channels , 2003, IEEE Trans. Commun..

[15]  Rüdiger L. Urbanke,et al.  Density Evolution, Thresholds and the Stability Condition for Non-binary LDPC Codes , 2005, ArXiv.

[16]  Masoud Salehi,et al.  Turbo codes and turbo coded modulation systems: analysis and performance bounds , 1998 .

[17]  Donald E. Knuth,et al.  The art of computer programming. Vol.2: Seminumerical algorithms , 1981 .

[18]  Donald Ervin Knuth,et al.  The Art of Computer Programming , 1968 .

[19]  Igal Sason,et al.  An Improved Sphere-Packing Bound for Finite-Length Codes Over Symmetric Memoryless Channels , 2006, IEEE Transactions on Information Theory.

[20]  C. Shannon.  Probability of error for optimal codes in a Gaussian channel , 1959 .

[21]  Gregory Poltyrev,et al.  Bounds on the decoding error probability of binary linear codes via their spectra , 1994, IEEE Trans. Inf. Theory.

[22]  Marc P. C. Fossorier,et al.  Sphere-packing bounds revisited for moderate block lengths , 2004, IEEE Transactions on Information Theory.

[23]  Dariush Divsalar,et al.  Upper bounds to error probabilities of coded systems beyond the cutoff rate , 2003, IEEE Trans. Commun..

[24]  Patrick Robertson,et al.  Bandwidth-Efficient Turbo Trellis-Coded Modulation Using Punctured Component Codes , 1998, IEEE J. Sel. Areas Commun..

[25]  David J. C. MacKay,et al.  Low-density parity check codes over GF(q) , 1998, IEEE Communications Letters.

[26]  Uri Erez,et al.  The ML decoding performance of LDPC ensembles over Z/sub q/ , 2005, IEEE Transactions on Information Theory.

[27]  Robert G. Gallager,et al.  A simple derivation of the coding theorem and some applications , 1965, IEEE Trans. Inf. Theory.

[28]  Emina Soljanin,et al.  Reliable channel regions for good binary codes transmitted over parallel channels , 2006, IEEE Transactions on Information Theory.

[29]  Ruoheng Liu,et al.  Adaptive Transmission with Variable-Rate Turbo Bit-Interleaved Coded Modulation , 2007, IEEE Transactions on Wireless Communications.

[30]  Robert F. H. Fischer,et al.  Multilevel codes: Theoretical concepts and practical design rules , 1999, IEEE Trans. Inf. Theory.

[31]  Shlomo Shamai,et al.  Variations on the Gallager bounds, connections, and applications , 2002, IEEE Trans. Inf. Theory.

[32]  H. Herzberg,et al.  Techniques of bounding the probability of decoding error for block coded modulation structures , 1994, IEEE Trans. Inf. Theory.

[33]  Meir Feder,et al.  Random coding techniques for nonrandom codes , 1999, IEEE Trans. Inf. Theory.

[34]  Shlomo Shamai,et al.  On improved bounds on the decoding error probability of block codes over interleaved fading channels, with applications to turbo-like codes , 2001, IEEE Trans. Inf. Theory.

[35]  Elwyn R. Berlekamp,et al.  Lower Bounds to Error Probability for Coding on Discrete Memoryless Channels. II , 1967, Inf. Control..

[36]  David Burshtein,et al.  Design and analysis of nonbinary LDPC codes for arbitrary discrete-memoryless channels , 2005, IEEE Transactions on Information Theory.

[37]  Idan Goldenberg,et al.  Coding for Parallel Channels: Gallager Bounds and Applications to Turbo-Like Codes , 2007, IEEE Transactions on Information Theory.

[38]  R. Gallager.  Information Theory and Reliable Communication , 1968 .

[39]  ShamaiShlomo,et al.  Performance analysis of linear codes under maximum-likelihood decoding , 2006 .

[40]  Dariush Divsalar,et al.  Bandwidth efficient parallel concatenated coding schemes , 1995 .

[41]  David Burshtein,et al.  Bounds on the maximum-likelihood decoding error probability of low-density parity-check codes , 2000, IEEE Trans. Inf. Theory.