Spatially Coupled LDPC Ensembles on Burst Erasure Channels

Spatially-Coupled LDPC (SC-LDPC) ensembles have gained significant interest since they were shown to universally achieve the capacity of binary memoryless channels under low-complexity belief-propagation decoding. In this work, we focus on the performance of these ensembles over binary erasure channels affected additionally by bursts. We assume that the burst can erase either a complete spatial position, modeling node failures in distributed storage, or can span across multiple spatial positions. We study the expected performance of random regular SC-LDPC ensembles on single-burst-erasure channels and provide tight lower bounds for the block erasure probability ($P_B$) at finite block length and bounds on the coupling parameter for being asymptotically able to recover the burst. Subsequently, we show the effect of introducing additional random erasures to these channels. Finally, we show that expurgation can bring substantial improvements in the block error rate by analyzing the minimal stopping sets of some expurgated code ensembles. Our study shows that for a fixed asymptotic code rate, the combination of increasing the variable node degree and expurgating the ensemble can improve the block error probability by several orders of magnitude. All the results are verified using Monte-Carlo simulations.

[1]  Alon Orlitsky,et al.  Stopping set distribution of LDPC code ensembles , 2003, IEEE Transactions on Information Theory.

[2]  Pablo M. Olmos,et al.  Scaling behavior of convolutional LDPC ensembles over the BEC , 2011, 2011 IEEE International Symposium on Information Theory Proceedings.

[3]  A. Orlitsky,et al.  Stopping sets and the girth of Tanner graphs , 2002, Proceedings IEEE International Symposium on Information Theory,.

[4]  Rüdiger L. Urbanke,et al.  Threshold Saturation via Spatial Coupling: Why Convolutional LDPC Ensembles Perform So Well over the BEC , 2010, IEEE Transactions on Information Theory.

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

[6]  Rüdiger L. Urbanke,et al.  Modern Coding Theory , 2008 .

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

[8]  Kamil Sh. Zigangirov,et al.  Time-varying periodic convolutional codes with low-density parity-check matrix , 1999, IEEE Trans. Inf. Theory.

[9]  Rüdiger L. Urbanke,et al.  Spatially coupled ensembles universally achieve capacity under belief propagation , 2012, 2012 IEEE International Symposium on Information Theory Proceedings.

[10]  Pablo M. Olmos,et al.  Analyzing finite-length protograph-based spatially coupled LDPC codes , 2014, 2014 IEEE International Symposium on Information Theory.

[11]  Gerhard Fettweis,et al.  Tail-Biting LDPC Convolutional Codes , 2007, 2007 IEEE International Symposium on Information Theory.

[12]  Paul H. Siegel,et al.  Protograph-Based LDPC Convolutional Codes for Correlated Erasure Channels , 2010, 2010 IEEE International Conference on Communications.

[13]  Ezio Biglieri,et al.  Low-Density Parity-Check Codes for Nonergodic Block-Fading Channels , 2007, IEEE Transactions on Information Theory.

[14]  Gerhard Fettweis,et al.  Improving code diversity on block-fading channels by spatial coupling , 2014, 2014 IEEE International Symposium on Information Theory.

[15]  Gerhard Fettweis,et al.  Protograph design for spatially-coupled codes to attain an arbitrary diversity order , 2015, 2015 IEEE Information Theory Workshop - Fall (ITW).

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

[17]  Joseph Jean Boutros,et al.  Spatial coupling of root-LDPC: Parity bits doping , 2015, 2015 22nd International Conference on Telecommunications (ICT).

[18]  Gerhard Fettweis,et al.  On the thresholds of generalized LDPC convolutional codes based on protographs , 2010, 2010 IEEE International Symposium on Information Theory.

[19]  Laurent Schmalen,et al.  Spatially Coupled LDPC codes affected by a single random burst of erasures , 2016, 2016 9th International Symposium on Turbo Codes and Iterative Information Processing (ISTC).

[20]  Joseph Jean Boutros,et al.  Spatial coupling for distributed storage and diversity applications , 2015, 2015 5th International Conference on Communications and Networking (COMNET).

[21]  Gerhard Fettweis,et al.  SC-LDPC codes over the block-fading channel: Robustness to a synchronisation offset , 2015, 2015 IEEE International Black Sea Conference on Communications and Networking (BlackSeaCom).

[22]  Iryna Andriyanova,et al.  Performance bounds for spatially-coupled LDPC codes over the block erasure channel , 2013, 2013 IEEE International Symposium on Information Theory.

[23]  Michael Lentmaier,et al.  Spatially-coupled random access on graphs , 2012, 2012 IEEE International Symposium on Information Theory Proceedings.

[24]  Pablo M. Olmos,et al.  A Scaling Law to Predict the Finite-Length Performance of Spatially-Coupled LDPC Codes , 2014, IEEE Transactions on Information Theory.

[25]  Tadashi Wadayama,et al.  Band Splitting Permutations for Spatially Coupled LDPC Codes Enhancing Burst Erasure Immunity , 2015, ArXiv.