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IEEE International Symposium on Information Theory (ISIT)

Lower bounds on the graphical complexity of finite-length LDPC codes

Igal Sason

Published Jun 1, 2009• 0 citations
Publisher

Abstract

This paper considers information-theoretic lower bounds on the graphical complexity of finite-length LDPC codes. It is assumed that the transmission of the codes takes place over a memoryless binary-input output-symmetric (MBIOS) channel, and the bounds are expressed as a function of the code performance and their achievable gap to capacity (either under ML decoding or any sub-optimal decoding algorithm). The lower bounds on the graphical complexity are compared to some explicit LDPC codes (or code ensembles), showing that these bounds are informative for considering the fundamental tradeoff which exists between the performance and graphical complexity of finite-length LDPC codes. This work relies on the full paper version.

DOI
10.1109/isit.2009.5205822

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This paper considers information-theoretic lower bounds on the graphical complexity of finite-length LDPC codes. It is assumed that the transmission of the codes takes place over a memoryless binary-input output-symmetric (MBIOS) channel, and the bounds are expressed as a function of the code performance and their achievable gap to capacity (either under ML decoding or any sub-optimal decoding algorithm). The lower bounds on the graphical complexity are compared to some explicit LDPC codes (or code ensembles), showing that these bounds are informative for considering the fundamental tradeoff which exists between the performance and graphical complexity of finite-length LDPC codes. This work relies on the full paper version.
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This paper considers the achievable rates and decoding complexity of low-density parity-check (LDPC) codes over statistically independent parallel channels. The paper starts with the derivation of bounds on the conditional entropy of the transmitted codeword given the received sequence at the output of the parallel channels; the component channels are considered to be memoryless, binary-input, and output-symmetric (MBIOS). These results serve for the derivation of an upper bound on the achievable rates of ensembles of LDPC codes under optimal maximum-likelihood (ML) decoding when their transmission takes place over parallel MBIOS channels. The paper relies on the latter bound for obtaining upper bounds on the achievable rates of ensembles of randomly and intentionally punctured LDPC codes over MBIOS channels. The paper also provides a lower bound on the decoding complexity (per iteration) of ensembles of LDPC codes under message-passing iterative decoding over parallel MBIOS channels; the bound is given in terms of the gap between the rate of these codes for which reliable communication is achievable and the channel capacity. The paper presents a diagram which shows interconnections between the theorems introduced in this paper and some other previously reported results. The setting which serves for the derivation of the bounds on the achievable rates and decoding complexity is general, and the bounds can be applied to other scenarios which can be treated as different forms of communication over parallel channels.
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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.