Non-Binary LDPC Erasure Codes With Separated Low-Degree Variable Nodes

The code design of non-binary low-density paritycheck codes for the erasure channel, under maximum a posteriori decoding, is addressed. In particular, a partially structured ensemble of codes, characterized by a careful control of the amount and of the connectivity of the variable nodes of small degrees, is proposed. The identified ensemble of codes is analyzed in terms of asymptotic thresholds and weight distribution and it is shown that codes from the ensemble provide a remarkable trade-off between waterfall performance, error floor, and decoding complexity. As an example, the performance curve of a short (256,128) code on the memoryless 16-ary erasure channel tightly approaches the Singleton bound at least down to a codeword error rate of 10-9, at low decoding complexity.

[1]  Andrea Montanari,et al.  Maxwell Construction: The Hidden Bridge Between Iterative and Maximum a Posteriori Decoding , 2005, IEEE Transactions on Information Theory.

[2]  Marco Chiani,et al.  Spectral shape of non-binary LDPC code ensembles with separated variable nodes , 2014, 2014 International Symposium on Information Theory and its Applications.

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

[4]  T. Richrdson,et al.  Finite-length analysis of various low-density parity-check ensembles for the binary erasure channel , 2002, Proceedings IEEE International Symposium on Information Theory,.

[5]  Igal Sason,et al.  Accumulate–Repeat–Accumulate Codes: Capacity-Achieving Ensembles of Systematic Codes for the Erasure Channel With Bounded Complexity , 2007, IEEE Transactions on Information Theory.

[6]  David Declercq,et al.  Weight Distributions of Non-binary LDPC Codes , 2011, IEICE Trans. Fundam. Electron. Commun. Comput. Sci..

[7]  Marco Chiani,et al.  Non-binary low-density parity-check codes for the q-ary erasure channel , 2013, 2013 IEEE International Conference on Communications (ICC).

[8]  Michael Luby,et al.  LT codes , 2002, The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings..

[9]  Marco Chiani,et al.  Performance versus overhead for fountain codes over Fq , 2010, IEEE Communications Letters.

[10]  T. Richardson,et al.  Multi-Edge Type LDPC Codes , 2004 .

[11]  David Declercq,et al.  Design of regular (2,d/sub c/)-LDPC codes over GF(q) using their binary images , 2008, IEEE Transactions on Communications.

[12]  Marco Chiani,et al.  Bounds on the Error Probability of Block Codes over the q-Ary Erasure Channel , 2013, IEEE Transactions on Communications.

[13]  Robert J. McEliece,et al.  Practical codes for photon communication , 1981, IEEE Trans. Inf. Theory.

[14]  Andrea Montanari,et al.  The Generalized Area Theorem and Some of its Consequences , 2005, IEEE Transactions on Information Theory.

[15]  Changyan Di.  Asymptotic and finite-length analysis of low-density parity-check codes , 2004 .

[16]  Sae-Young Chung,et al.  An efficient algorithm for ML decoding of raptor codes over the binary erasure channel , 2008, IEEE Communications Letters.

[17]  Muriel Médard,et al.  On coding for reliable communication over packet networks , 2005, Phys. Commun..

[18]  Milica Stojanovic,et al.  On Coding for Delay—Network Coding for Time-Division Duplexing , 2012, IEEE Transactions on Information Theory.

[19]  Marco Chiani,et al.  Maximum Likelihood Erasure Decoding of LDPC Codes: Pivoting Algorithms and Code Design , 2012, IEEE Transactions on Communications.

[20]  Peter Vary,et al.  The performance of low-density random linear fountain codes over higher order galois fields under maximum likelihood decoding , 2011, 2011 49th Annual Allerton Conference on Communication, Control, and Computing (Allerton).

[21]  Jörg Kliewer,et al.  Joint design of channel and network coding for star networks connected by binary symmetric channels , 2013, IEEE Transactions on Communications.

[22]  Radford M. Neal,et al.  Near Shannon limit performance of low density parity check codes , 1996 .

[23]  Luigi Rizzo,et al.  Effective erasure codes for reliable computer communication protocols , 1997, CCRV.

[24]  David Burshtein,et al.  Efficient maximum-likelihood decoding of LDPC codes over the binary erasure channel , 2004, IEEE Transactions on Information Theory.

[25]  Rüdiger L. Urbanke,et al.  Efficient encoding of low-density parity-check codes , 2001, IEEE Trans. Inf. Theory.

[26]  Daniel A. Spielman,et al.  Practical loss-resilient codes , 1997, STOC '97.

[27]  Stephan ten Brink,et al.  Extrinsic information transfer functions: model and erasure channel properties , 2004, IEEE Transactions on Information Theory.

[28]  Amin Shokrollahi,et al.  Capacity-achieving sequences for the erasure channel , 2002, IEEE Trans. Inf. Theory.

[29]  J. L. Massey,et al.  Capacity, Cutoff Rate, and Coding for a Direct-Detection Optical Channel , 1981, IEEE Trans. Commun..

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

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

[32]  Faramarz Fekri,et al.  On decoding of low-density parity-check codes over the binary erasure channel , 2004, IEEE Transactions on Information Theory.

[33]  Marco Chiani,et al.  A robust pulse position coded modulation scheme for the Poisson channel , 2014, 2014 IEEE International Conference on Communications (ICC).

[34]  Daniel A. Spielman,et al.  Efficient erasure correcting codes , 2001, IEEE Trans. Inf. Theory.

[35]  E.R. Berlekamp,et al.  The technology of error-correcting codes , 1980, Proceedings of the IEEE.