On Universal Properties of Capacity-Approaching LDPC Ensembles
📝 Abstract
This paper is focused on the derivation of some universal properties of capacity-approaching low-density parity-check (LDPC) code ensembles whose transmission takes place over memoryless binary-input output-symmetric (MBIOS) channels. Properties of the degree distributions, graphical complexity and the number of fundamental cycles in the bipartite graphs are considered via the derivation of information-theoretic bounds. These bounds are expressed in terms of the target block/ bit error probability and the gap (in rate) to capacity. Most of the bounds are general for any decoding algorithm, and some others are proved under belief propagation (BP) decoding. Proving these bounds under a certain decoding algorithm, validates them automatically also under any sub-optimal decoding algorithm. A proper modification of these bounds makes them universal for the set of all MBIOS channels which exhibit a given capacity. Bounds on the degree distributions and graphical complexity apply to finite-length LDPC codes and to the asymptotic case of an infinite block length. The bounds are compared with capacity-approaching LDPC code ensembles under BP decoding, and they are shown to be informative and are easy to calculate. Finally, some interesting open problems are considered.
💡 Analysis
This paper is focused on the derivation of some universal properties of capacity-approaching low-density parity-check (LDPC) code ensembles whose transmission takes place over memoryless binary-input output-symmetric (MBIOS) channels. Properties of the degree distributions, graphical complexity and the number of fundamental cycles in the bipartite graphs are considered via the derivation of information-theoretic bounds. These bounds are expressed in terms of the target block/ bit error probability and the gap (in rate) to capacity. Most of the bounds are general for any decoding algorithm, and some others are proved under belief propagation (BP) decoding. Proving these bounds under a certain decoding algorithm, validates them automatically also under any sub-optimal decoding algorithm. A proper modification of these bounds makes them universal for the set of all MBIOS channels which exhibit a given capacity. Bounds on the degree distributions and graphical complexity apply to finite-length LDPC codes and to the asymptotic case of an infinite block length. The bounds are compared with capacity-approaching LDPC code ensembles under BP decoding, and they are shown to be informative and are easy to calculate. Finally, some interesting open problems are considered.
📄 Content
Low-density parity-check (LDPC) codes form a class of powerful error-correcting codes which are efficiently encoded and decoded with low-complexity algorithms. These linear block codes, originally introduced by Gallager in the early sixties [14], are characterized by sparse paritycheck matrices which facilitate their low-complexity decoding with iterative message-passing algorithms. In spite of the seminal work of Gallager, LDPC codes were ignored for a long time. Following the breakthrough in coding theory, made by the introduction of turbo codes [5] and the rediscovery of LDPC codes [25] in the mid 1990s, it was realized that an efficient design of these codes enables to closely approach the channel capacity while maintaining reasonable decoding complexity. This breakthrough attracted many coding-theorists during the last decade (see, e.g., [9], [37], [55]).
The asymptotic analysis of LDPC code ensembles under iterative message-passing decoding algorithms relies on the density evolution (DE) approach which was developed by Richardson and Urbanke (see [34], [35], [37]). This technique is commonly used for optimizing the degree distributions of capacity-approaching LDPC code ensembles where the target is to maximize the achievable rate for a given channel model or to maximize the threshold for a given code rate subject to some constraints on the degree distributions [2]. Some approximate techniques which optimize the degree distributions of LDPC code ensembles under further practical constraints are of interest (e.g., an optimization for obtaining a good tradeoff between the asymptotic gap to capacity and the decoding complexity [3]). For the binary erasure channel (BEC), the DE approach is much simplified since it leads to a one-dimensional analysis. As a result of this significant simplification, some explicit expressions for capacity-achieving sequences of LDPC code ensembles have been derived for the BEC (see, e.g., [24], [29], [37] and [48]). For general memoryless binary-input output-symmetric (MBIOS) channels, as of yet there are no closedform expressions for capacity-achieving LDPC code ensembles under iterative decoding, and the DE technique serves as a numerical tool for the design of capacity-approaching LDPC code ensembles in the limit where their block length tends to infinity. Although maximum-likelihood (ML) decoding is prohibitively complex, capacity-achieving sequences of LDPC code ensembles have been constructed under ML decoding for any MBIOS channel where the analysis relies on upper bounds on the decoding error probability which are based on the distance spectra of these ensembles (see [18], [19], [39], and [40,Theorem 2.2]).
Consider right-regular LDPC codes (i.e., LDPC codes where the degree of the parity-check nodes is fixed to a certain value a R ), and assume that their transmission takes place over a binary symmetric channel (BSC). In his thesis, Gallager derived an upper bound on the maximal achievable rate of these codes where it is required to obtain vanishing block error probability as we let the block length tend to infinity (see [14,Theorem 3.3]). This information-theoretic bound holds under ML decoding or any sub-optimal decoding algorithm. This bound shows that rightregular LDPC codes cannot achieve the channel capacity on a BSC, even under ML decoding. Based on this bound, the inherent gap between the achievable rate and the channel capacity is well approximated by an expression which decreases to zero exponentially fast in a R . Burshtein et al. have generalized Gallager’s bound for general LDPC code ensembles whose transmission takes place over an MBIOS channel [7]. An improved upper bound on the achievable rates of LDPC code ensembles was obtained by Wiechman and Sason [53], followed by a generalization of this bound to the case where the transmission takes place over a set of parallel MBIOS channels [41]. This work partially relies on the analysis in [53] (see Section II for relevant background). Khandekar and McEliece suggested to measure the encoding and decoding complexity of codes defined on graphs in terms of the achievable gap (in rate) to capacity, and they also had some conjectures regarding the behavior of the complexity as the gap to capacity vanishes [21]. Following their approach, the tradeoff between the performance and complexity is analyzed in the literature for LDPC code ensembles and some other variants of codes defined on graphs (see, e.g., [18], [19], [31], [32], [40], [41], [42], [53] and references therein).
In this paper, we consider some properties of capacity-approaching LDPC code ensembles whose transmission takes place over MBIOS channels. One question which is addressed in this paper is the following: Question 1: How do the degree distributions of capacity-approaching LDPC code ensembles behave as a function of the achievable gap (in rate) to capacity ?
The behavior of the degree distributions of capacity-approaching LDPC code ensembles
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