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Journal of Communications Technology and Electronics

Estimation of the exponent of the decoding error probability for a special generalized LDPC code

Victor Zyablov, Pavel S. Rybin

Published Aug 1, 2012• 4 citations
Publisher

Abstract

A special construction of a generalized low-density parity-check (LDPC) code and a low-complexity algorithm for his code decoding are proposed. A lower estimate of the exponent of the decoding error probability is obtained for the considered code and the decoding algorithm. This estimate leads the conclusion that, in an ensemble of considered LDPC codes, there are codes with rates as high as the code capacity and the exponent of the decoding error probability exceeds zero.

DOI
10.1134/s1064226912080098

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A special construction of a generalized low-density parity-check (LDPC) code and a low-complexity algorithm for his code decoding are proposed. A lower estimate of the exponent of the decoding error probability is obtained for the considered code and the decoding algorithm. This estimate leads the conclusion that, in an ensemble of considered LDPC codes, there are codes with rates as high as the code capacity and the exponent of the decoding error probability exceeds zero.
Two ensembles of low-density parity-check (LDPC) codes with low-complexity decoding algorithms are considered. The first ensemble consists of generalized LDPC codes, and the second consists of concatenated codes with an outer LDPC code. Error exponent lower bounds for these ensembles under the corresponding low-complexity decoding algorithms are compared. A modification of the decoding algorithm of a generalized LDPC code with a special construction is proposed. The error exponent lower bound for the modified decoding algorithm is obtained. Finally, numerical results for the considered error exponent lower bounds are presented and analyzed.
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.
This paper deals with the specific construction of binary low-density parity-check (LDPC) codes. We derive lower bounds on the error exponents for these codes transmitted over the memoryless binary symmetric channel (BSC) for both the well-known maximum-likelihood (ML) and proposed low-complexity decoding algorithms. We prove the existence of such LDPC codes that the probability of erroneous decoding decreases exponentially with the growth of the code length while keeping coding rates below the corresponding channel capacity. We also show that an obtained error exponent lower bound under ML decoding almost coincide with the error exponents of good linear codes.
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.
This paper deals with the method of undetected error probability estimation for a low-density parity-check (LDPC) code under any given iterative decoding algorithm. We propose such modification of a given iterative decoding algorithm, that almost preserves a decoding failure exponent and decoding complexity of this algorithm. We obtain the upper bound on the undetected error probability for the modified algorithm. We show how to use the proposed method to estimate the undetected error probability of LDPC code under the belief propagation (BP) algorithm at the end of this paper.
We derive both upper and lower bounds on the decoding error probability of maximum-likelihood (ML) decoded low-density parity-check (LDPC) codes. The results hold for any binary-input symmetric-output channel. Our results indicate that for various appropriately chosen ensembles of LDPC codes, reliable communication is possible up to channel capacity. However, the ensemble averaged decoding error probability decreases polynomially, and not exponentially. The lower and upper bounds coincide asymptotically, thus showing the tightness of the bounds. However, for ensembles with suitably chosen parameters, the error probability of almost all codes is exponentially decreasing, with an error exponent that can be set arbitrarily close to the standard random coding exponent.