Error Floor Approximation for LDPC Codes in the AWGN Channel

This paper addresses the prediction of error floors of low-density parity-check codes transmitted over the additive white Gaussian noise channel. Using a linear state-space model to estimate the behavior of the sum-product algorithm (SPA) decoder in the vicinity of trapping sets (TSs), we study the performance of the SPA decoder in the log-likelihood ratio (LLR) domain as a function of the LLR saturation level. When applied to several widely studied codes, the model accurately predicts a significant decrease in the error floor as the saturation level is allowed to increase. For nonsaturating decoders, however, we find that the state-space model breaks down after a small number of iterations due to the strong correlation of LLR messages. We then revisit Richardson's importance-sampling methodology for estimating error floors due to TSs when those floors are too low for Monte Carlo simulation. We propose modifications that account for the behavior of a nonsaturating decoder and present the resulting error floor estimates for the Margulis code. These estimates are much lower, significantly steeper, and more sensitive to iteration count than those previously reported.

[1]  Richard A. Brualdi,et al.  Combinatorial matrix theory , 1991 .

[2]  Daniel J. Costello,et al.  LDPC block and convolutional codes based on circulant matrices , 2004, IEEE Transactions on Information Theory.

[3]  Hans-Andrea Loeliger,et al.  Codes and iterative decoding on general graphs , 1995, Eur. Trans. Telecommun..

[4]  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.

[5]  Paul H. Siegel,et al.  Quantized Iterative Message Passing Decoders with Low Error Floor for LDPC Codes , 2012, IEEE Transactions on Communications.

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

[7]  David Declercq,et al.  Finite alphabet iterative decoding (FAID) of the (155,64,20) Tanner code , 2010, 2010 6th International Symposium on Turbo Codes & Iterative Information Processing.

[8]  O. Milenkovic,et al.  Algorithmic and combinatorial analysis of trapping sets in structured LDPC codes , 2005, 2005 International Conference on Wireless Networks, Communications and Mobile Computing.

[9]  William E. Ryan,et al.  Enumerators for Protograph-Based Ensembles of LDPC and Generalized LDPC Codes , 2011, IEEE Transactions on Information Theory.

[10]  Martin J. Wainwright,et al.  An Efficient 10GBASE-T Ethernet LDPC Decoder Design With Low Error Floors , 2010, IEEE Journal of Solid-State Circuits.

[11]  Shu Lin,et al.  A class of low-density parity-check codes constructed based on Reed-Solomon codes with two information symbols , 2003, IEEE Communications Letters.

[12]  Lara Dolecek,et al.  Analysis of Absorbing Sets and Fully Absorbing Sets of Array-Based LDPC Codes , 2009, IEEE Transactions on Information Theory.

[13]  Jing Sun.  Studies on graph-based coding systems , 2004 .

[14]  Jon Hamkins.  Performance of low-density parity-check coded modulation , 2010, 2010 IEEE Aerospace Conference.

[15]  Johannes B. Huber,et al.  The Trapping Redundancy of Linear Block Codes , 2006, IEEE Transactions on Information Theory.

[16]  H. Vincent Poor,et al.  Finding All Small Error-Prone Substructures in LDPC Codes , 2009, IEEE Transactions on Information Theory.

[17]  Simon Litsyn,et al.  Decreasing error floor in LDPC codes by parity-check matrix extensions , 2009, 2009 IEEE International Symposium on Information Theory.

[18]  Shashi Kiran Chilappagari,et al.  On the Construction of Structured LDPC Codes Free of Small Trapping Sets , 2012, IEEE Transactions on Information Theory.

[19]  Stark C. Draper,et al.  Divide and Concur and Difference-Map BP Decoders for LDPC Codes , 2011, IEEE Transactions on Information Theory.

[20]  David Declercq,et al.  Trapping set enumerators for specific LDPC codes , 2010, 2010 Information Theory and Applications Workshop (ITA).

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

[22]  Richard D. Wesel,et al.  LDPC Decoders with Informed Dynamic Scheduling , 2010, IEEE Transactions on Communications.

[23]  Hua Xiao,et al.  Error Rate Estimation of Low-Density Parity-Check Codes Decoded by Quantized Soft-Decision Iterative Algorithms , 2013, IEEE Transactions on Communications.

[24]  Lara Dolecek,et al.  Predicting error floors of structured LDPC codes: deterministic bounds and estimates , 2009, IEEE Journal on Selected Areas in Communications.

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

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

[27]  William Ryan,et al.  Channel Codes: Classical and Modern , 2009 .

[28]  Paul H. Siegel,et al.  LDPC Code Density Evolution in the Error Floor Region , 2014, ArXiv.

[29]  Jinghu Chen,et al.  Density evolution for BP-based decoding algorithms of LDPC codes and their quantized versions , 2002, Global Telecommunications Conference, 2002. GLOBECOM '02. IEEE.

[30]  Paul H. Siegel,et al.  Efficient Algorithms to Find All Small Error-Prone Substructures in LDPC Codes , 2011, 2011 IEEE Global Telecommunications Conference - GLOBECOM 2011.

[31]  Paul H. Siegel,et al.  Numerical issues affecting LDPC error floors , 2012, 2012 IEEE Global Communications Conference (GLOBECOM).

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

[33]  Michael E. O'Sullivan,et al.  The Sum-Product Algorithm for Degree-2 Check Nodes and Trapping Sets , 2014, ArXiv.

[34]  Shuai Zhang,et al.  On the dynamics of the error floor behavior in regular LDPC codes , 2009, 2009 IEEE Information Theory Workshop.

[35]  Alexander Vardy,et al.  Which codes have cycle-free Tanner graphs? , 1999, IEEE Trans. Inf. Theory.

[36]  Stephen G. Wilson,et al.  A General Method for Finding Low Error Rates of LDPC Codes , 2006, ArXiv.

[37]  Minyue Fu.  On Gaussian Approximation for Density Evolution of Low-Density Parity-Check Codes , 2006, 2006 IEEE International Conference on Communications.

[38]  Shuai Zhang,et al.  Controlling the Error Floor in LDPC Decoding , 2012, IEEE Transactions on Communications.

[39]  Lara Dolecek,et al.  Design of LDPC decoders for improved low error rate performance: quantization and algorithm choices , 2009, IEEE Transactions on Communications.

[40]  Yifei Zhang,et al.  Toward low LDPC-code floors: a case study , 2009, IEEE Transactions on Communications.

[41]  Amir H. Banihashemi,et al.  An efficient algorithm for finding dominant trapping sets of LDPC codes , 2011, 2010 6th International Symposium on Turbo Codes & Iterative Information Processing.

[42]  Martin Bossert,et al.  Optimization of a reduced-complexity decoding algorithm for LDPC codes by density evolution , 2005, IEEE International Conference on Communications, 2005. ICC 2005. 2005.

[43]  Emina Soljanin,et al.  Asymptotic Spectra of Trapping Sets in Regular and Irregular LDPC Code Ensembles , 2007, IEEE Transactions on Information Theory.

[44]  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.

[45]  William Ryan,et al.  Channel Codes by William Ryan , 2009 .

[46]  Achilleas Anastasopoulos,et al.  A comparison between the sum-product and the min-sum iterative detection algorithms based on density evolution , 2001, GLOBECOM'01. IEEE Global Telecommunications Conference (Cat. No.01CH37270).

[47]  William E. Ryan,et al.  Low-floor decoders for LDPC codes , 2009, IEEE Transactions on Communications.

[48]  Ajay Dholakia,et al.  Reduced-complexity decoding of LDPC codes , 2005, IEEE Transactions on Communications.

[49]  Amir H. Banihashemi,et al.  On implementation of min-sum algorithm and its modifications for decoding low-density Parity-check (LDPC) codes , 2005, IEEE Transactions on Communications.

[50]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[51]  Carl D. Meyer,et al.  Matrix Analysis and Applied Linear Algebra , 2000 .