Finite-Dimensional Bounds on ${\BBZ}_m$ and Binary LDPC Codes With Belief Propagation Decoders

This paper focuses on finite-dimensional upper and lower bounds on decodable thresholds of Zopfm and binary low-density parity-check (LDPC) codes, assuming belief propagation decoding on memoryless channels. A concrete framework is presented, admitting systematic searches for new bounds. Two noise measures are considered: the Bhattacharyya noise parameter and the soft bit value for a maximum a posteriori probability (MAP) decoder on the uncoded channel. For Zopf m LDPC codes, an iterative m-dimensional bound is derived for m-ary-input/symmetric-output channels, which gives a sufficient stability condition for Zopfm LDPC codes and is complemented by a matched necessary stability condition introduced herein. Applications to coded modulation and to codes with nonequiprobably distributed codewords are also discussed. For binary codes, two new lower bounds are provided for symmetric channels, including a two-dimensional iterative bound and a one-dimensional noniterative bound, the latter of which is the best known bound that is tight for binary-symmetric channels (BSCs), and is a strict improvement over the existing bound derived by the channel degradation argument. By adopting the reverse channel perspective, upper and lower bounds on the decodable Bhattacharyya noise parameter are derived for nonsymmetric channels, which coincides with the existing bound for symmetric channels

[1]  Sae-Young Chung,et al.  On the design of low-density parity-check codes within 0.0045 dB of the Shannon limit , 2001, IEEE Communications Letters.

[2]  W. Freeman,et al.  Bethe free energy, Kikuchi approximations, and belief propagation algorithms , 2001 .

[3]  Alain Glavieux,et al.  Reflections on the Prize Paper : "Near optimum error-correcting coding and decoding: turbo codes" , 1998 .

[4]  Hans-Andrea Loeliger,et al.  Signal sets matched to groups , 1991, IEEE Trans. Inf. Theory.

[5]  Robert J. McEliece.  Are Turbo-like Codes Effective on Nonstandard Channels?* , 2001 .

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

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

[8]  Ge Li,et al.  Analysis of nonbinary ldpc codes using gaussian approximation , 2003, IEEE International Symposium on Information Theory, 2003. Proceedings..

[9]  D. Burshtein,et al.  Bounds on the performance of belief propagation decoding , 2001, Proceedings. 2001 IEEE International Symposium on Information Theory (IEEE Cat. No.01CH37252).

[10]  Dariush Divsalar,et al.  Coding theorems for 'turbo-like' codes , 1998 .

[11]  H. Vincent Poor,et al.  Density evolution for asymmetric memoryless channels , 2005, IEEE Transactions on Information Theory.

[12]  A. Hasman,et al.  Probabilistic reasoning in intelligent systems: Networks of plausible inference , 1991 .

[13]  Kenta Kasai,et al.  Detailed representation of irregular LDPC code ensembles and density evolution , 2003, IEEE International Symposium on Information Theory, 2003. Proceedings..

[14]  F. Burkert,et al.  Turbo-decoding of nonbinary codes , 1997, Proceedings of IEEE International Symposium on Information Theory.

[15]  Xiao Ma,et al.  Binary intersymbol interference channels: Gallager codes, density evolution, and code performance bounds , 2003, IEEE Transactions on Information Theory.

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

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

[18]  Sae-Young Chung,et al.  On the capacity of low-density parity-check codes , 2001, Proceedings. 2001 IEEE International Symposium on Information Theory (IEEE Cat. No.01CH37252).

[19]  Robert J. McEliece,et al.  RA Codes Achieve AWGN Channel Capacity , 1999, AAECC.

[20]  Rüdiger L. Urbanke,et al.  Exact thresholds and optimal codes for the binary-symmetric channel and Gallager's decoding algorithm A , 2000, IEEE Transactions on Information Theory.

[21]  Johannes B. Huber,et al.  Bounds on information combining , 2005, IEEE Transactions on Information Theory.

[22]  D. Mackay,et al.  Low-Density Parity Check Codes over , 1998 .

[23]  Shlomo Shamai,et al.  Extremes of information combining , 2005, IEEE Transactions on Information Theory.

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

[25]  J. Berkmann.  A symbol-by-symbol MAP decoding rule for linear codes over rings using the dual code , 1998, Proceedings. 1998 IEEE International Symposium on Information Theory (Cat. No.98CH36252).

[26]  Sae-Young Chung,et al.  Analysis of sum-product decoding of low-density parity-check codes using a Gaussian approximation , 2001, IEEE Trans. Inf. Theory.

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

[28]  Giuseppe Caire,et al.  Bit-Interleaved Coded Modulation , 2008, Found. Trends Commun. Inf. Theory.

[29]  David Burshtein,et al.  Bounds on the performance of belief propagation decoding , 2002, IEEE Trans. Inf. Theory.

[30]  Stephan ten Brink,et al.  Convergence behavior of iteratively decoded parallel concatenated codes , 2001, IEEE Trans. Commun..

[31]  Sae-Young Chung,et al.  On the construction of some capacity-approaching coding schemes , 2000 .

[32]  U. Erez,et al.  The ML decoding performance of LDPC ensembles over /spl Zopf//sub q/ , 2003, IEEE International Symposium on Information Theory, 2003. Proceedings..

[33]  David Burshtein,et al.  Iterative Decoding of LDPC Codes over Arbitrary Discrete-Memoryless Channels , 2003 .

[34]  David J. C. MacKay,et al.  Sparse low-density parity-check codes for channels with cross-talk , 2003, Proceedings 2003 IEEE Information Theory Workshop (Cat. No.03EX674).