Finite-Length Scaling for Iteratively Decoded LDPC Ensembles

We investigate the behavior of iteratively decoded low-density parity-check (LDPC) codes over the binary erasure channel in the so-called ldquowaterfall region.rdquo We show that the performance curves in this region follow a simple scaling law. We conjecture that essentially the same scaling behavior applies in a much more general setting and we provide some empirical evidence to support this conjecture. The scaling law, together with the error floor expressions developed previously, can be used for a fast finite-length optimization.

[1]  I. Andriyanova.  Finite-Length Scaling of Repeat-Accumulate Codes on the BEC , 2008, 2008 IEEE International Zurich Seminar on Communications.

[2]  L. Schulman.  In: Finite size scaling and numerical simulation of statistical systems , 1990 .

[3]  Andrea Montanari,et al.  Finite-Length Scaling and Finite-Length Shift for Low-Density Parity-Check Codes , 2004, ArXiv.

[4]  C. McDiarmid.  Concentration , 1862, The Dental register.

[5]  MontanariAndrea,et al.  Finite-length scaling for iteratively decoded LDPC ensembles , 2009 .

[6]  P. Groeneboom.  Brownian motion with a parabolic drift and airy functions , 1989 .

[7]  V. F. Kolchin,et al.  Random Graphs: Contents , 1998 .

[8]  Gérard D. Cohen,et al.  The threshold probability of a code , 1995, IEEE Trans. Inf. Theory.

[9]  R. Urbanke,et al.  Weight Distribution of Iterative Coding Systems: How Deviant can You be? , 2001 .

[10]  Henry D. Pfister.  Finite-length analysis of a capacity-achieving ensemble for the binary erasure channel , 2005, IEEE Information Theory Workshop, 2005..

[11]  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,.

[12]  Andrea Montanari,et al.  The slope scaling parameter for general channels, decoders, and ensembles , 2008, 2008 IEEE International Symposium on Information Theory.

[13]  A. Dembo,et al.  Finite size scaling for the core of large random hypergraphs , 2007, math/0702007.

[14]  Alan M. Frieze,et al.  Random graphs , 2006, SODA '06.

[15]  R. Blahut,et al.  Bit Error Rate Estimate of Finite Length Turbo Codes , 2003 .

[16]  R. Stephenson.  A and V , 1962, The British journal of ophthalmology.

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

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

[19]  V. Rittenberg.  Finite-Size Scaling Theory , 1983 .

[20]  Gilles Zémor,et al.  Discrete Isoperimetric Inequalities and the Probability of a Decoding Error , 2000, Combinatorics, Probability and Computing.

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

[22]  V. E. Britikov.  Asymptotic number of forests from unrooted trees , 1988 .

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

[24]  R. E. Mills,et al.  Critical phenomena. , 1971, Science.

[25]  M. Freidlin,et al.  Random Perturbations of Dynamical Systems , 1984 .

[26]  Iryna Andriyanova.  Finite-length scaling of parallel turbo codes on the BEC , 2008, 2008 IEEE International Symposium on Information Theory.

[27]  Alon Orlitsky,et al.  Finite-length analysis of LDPC codes with large left degrees , 2002, Proceedings IEEE International Symposium on Information Theory,.

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

[29]  M. Habib.  Probabilistic methods for algorithmic discrete mathematics , 1998 .

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

[31]  A. Montanari.  The glassy phase of Gallager codes , 2001, cond-mat/0104079.

[32]  Andrea Montanari,et al.  How to find good finite-length codes: from art towards science , 2006, Eur. Trans. Telecommun..

[33]  Andrea Montanari,et al.  A Generalization of the Finite-Length Scaling Approach Beyond the BEC , 2007, 2007 IEEE International Symposium on Information Theory.