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Journal of Communications Technology and Electronics

On Estimation of the Error Exponent for Finite Length Regular Graph-Based LDPC Codes

Pavel S. Rybin, Fedor I. Ivanov

Published Dec 1, 2018• 1 citation
Publisher

Abstract

Abstract The error exponent of the regular graph-based binary low-density parity-check (LDPC) codes under the maximum likelihood (ML) decoding algorithm in the binary symmetric channel (BSC) is analyzed. Unlike most other papers where error exponents are considered for the case when the length of LDPC codes tends to infinity (asymptotic analysis), the finite length case (finite length analysis) is considered. In this paper, a method of deriving the lower bound on the error exponent for a regular graph-based LDPC code with finite length under ML decoding is described. Also we analyze Dependences of the error exponent on various LDPC code parameters are also analyzed. The numerical results obtained for the considered lower bound are represented and analyzed at the end of the paper.

DOI
10.1134/s1064226918120197

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In this paper we consider low-density parity-check (LDPC) codes with special construction. We obtain the lower-bounds on the error exponents for these codes under proposed low-complexity decoding algorithm and under a well known maximum likelihood decoding algorithm. We show that such LDPC code with special construction exists, that the error probability of the low-complexity decoding algorithm exponentially decreases with the code length for all code rates below the channel capacity. We also show that obtained lower-bound on the error exponent under the maximum likelihood decoding almost reaches the lower-bound on the error exponent of good linear codes under the maximum likelihood decoding. The error exponents are computed numerically for different code parameters.
We derive bounds on the error probability of ML decoded LDPC (low density parity check) codes, for any binary-input symmetric-output channel. For appropriately chosen ensembles of LDPC codes, reliable communication is possible up to channel capacity. The lower and upper bounds coincide asymptotically, indicating a polynomially decreasing ensemble averaged error probability. For ensembles with suitably chosen parameters, the error probability of almost all codes is exponentially decreasing. Furthermore, the error exponent can be set arbitrarily close to the standard random coding exponent.
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