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IEEE Transactions on Information Theory

Performance of low-density parity-check codes with linear minimum distance

Hossein Pishro-Nik, Faramarz Fekri

Published Dec 28, 2005• 13 citations
Publisher

Abstract

This correspondence studies the performance of the iterative decoding of low-density parity-check (LDPC) code ensembles that have linear typical minimum distance and stopping set size. We first obtain a lower bound on the achievable rates of these ensembles over memoryless binary-input output-symmetric channels. We improve this bound for the binary erasure channel. We also introduce a method to construct the codes meeting the lower bound for the binary erasure channel. Then, we give upper bounds on the rate of LDPC codes with linear minimum distance when their right degree distribution is fixed. We compare these bounds to the previously derived upper bounds on the rate when there is no restriction on the code ensemble.

DOI
10.1109/tit.2005.860439

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This correspondence studies the performance of the iterative decoding of low-density parity-check (LDPC) code ensembles that have linear typical minimum distance and stopping set size. We first obtain a lower bound on the achievable rates of these ensembles over memoryless binary-input output-symmetric channels. We improve this bound for the binary erasure channel. We also introduce a method to construct the codes meeting the lower bound for the binary erasure channel. Then, we give upper bounds on the rate of LDPC codes with linear minimum distance when their right degree distribution is fixed. We compare these bounds to the previously derived upper bounds on the rate when there is no restriction on the code ensemble.
We derive lower bounds on the density of parity-check matrices of binary linear codes which are used over memoryless binary-input output-symmetric (MBIOS) channels. The bounds are expressed in terms of the gap between the rate of these codes for which reliable communications is achievable and the channel capacity; they are valid for every sequence of binary linear block codes if there exists a decoding algorithm under which the average bit-error probability vanishes. For every MBIOS channel, we construct a sequence of ensembles of regular low-density parity-check (LDPC) codes, so that an upper bound on the asymptotic density of their parity-check matrices scales similarly to the lower bound. The tightness of the lower bound is demonstrated for the binary erasure channel by analyzing a sequence of ensembles of right-regular LDPC codes which was introduced by Shokrollahi, and which is known to achieve the capacity of this channel. Under iterative message-passing decoding, we show that this sequence of ensembles is asymptotically optimal (in a sense to be defined in this paper), strengthening a result of Shokrollahi. Finally, we derive lower bounds on the bit-error probability and on the gap to capacity for binary linear block codes which are represented by bipartite graphs, and study their performance limitations over MBIOS channels. The latter bounds provide a quantitative measure for the number of cycles of bipartite graphs which represent good error-correction codes.
The performance of maximum-likelihood (ML) decoding on the binary erasure channel for finite-length low-density parity-check (LDPC) codes from two random ensembles is studied. A tightened union-type upper bound on the ML decoding error probability based on the precise coefficients of the average weight spectrum is presented. For LDPC codes from the Gallager ensemble and the Richardson-Urbanke ensemble, new upper bounds on the ML decoding performance based on computing the rank of submatrices of the code parity-check matrix are derived. A new lower bound on the ML decoding threshold followed from the latter error probability bound is obtained. An improved lower bound on the error probability for codes with a known estimate on the minimum distance is presented as well. A new low-complexity near-ML decoding algorithm for quasi-cyclic LDPC codes is proposed and simulated. Its performance is compared to the simulated belief propagation and ML decoding performance and simulated performance of the best known improved iterative decoding techniques, as well as, with the derived upper bounds on the ML decoding performance and with decoding thresholds obtained by the density evolution technique.
A variety of communication scenarios can be modeled by a set of parallel channels. Upper bounds on the achievable rates under maximum-likelihood (ML) decoding, and lower bounds on the decoding complexity per iteration of ensembles of low-density parity-check (LDPC) codes are presented. The communication of these codes is assumed to take place over statistically independent parallel channels where the component channels are memoryless, binary-input, and output-symmetric. The bounds are applied to ensembles of punctured LDPC codes where the puncturing patterns are either random or possess some structure. Our discussion is concluded by a diagram showing interconnections between the new theorems and some previously reported results.