An Efficient Instanton Search Algorithm for LP Decoding of LDPC Codes Over the BSC

Authors:
S. K. Chilappagari;M. Chertkov;B. Vasic
Affiliations:
Dept. of Electr. & Comput. Eng., Univ. of Arizona, Tucson, AZ, USA;-;-
Venue:
IEEE Transactions on Information Theory
Year:
2011

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Analysis of error floors of LDPC codes under LP decoding over the BSC

ISIT'09 Proceedings of the 2009 IEEE international conference on Symposium on Information Theory - Volume 1

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Abstract

We consider linear programming (LP) decoding of a fixed low-density parity-check (LDPC) code over the binary symmetric channel (BSC). The LP decoder fails when it outputs a pseudo-codeword which is not equal to the transmitted codeword. We design an efficient algorithm termed the Instanton Search Algorithm (ISA) which generates an error vector called the BSC-instanton. We prove that: (a) the LP decoder fails for any error pattern with support that is a superset of the support of an instanton; (b) for any input, the ISA outputs an instanton in the number of steps upper-bounded by twice the number of errors in the input error vector. We then find the number of unique instantons of different sizes for a given LDPC code by running the ISA sufficient number of times.