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Conference on Information Sciences and Systems

Decoding complexity of irregular LDGM-LDPC codes over the BISOM channels

Manik Raina, Predrag Spasojević

Published Mar 1, 2010• 1 citation
Publisher

Abstract

An irregular LDGM-LDPC code is studied as a sub-code of an LDPC code with some randomly punctured output-bits. It is shown that the LDGM-LDPC codes achieve rates arbitrarily close to the channel-capacity of the binary-input symmetric-output memoryless (BISOM) channel with bounded complexity. The measure of complexity is the average-degree (per information-bit) of the check-nodes for the factor-graph of the code. A lower-bound on the average degree of the check-nodes of the irregular LDGM-LDPC codes is obtained. The bound does not depend on the decoder used at the receiver. The stability condition for decoding the irregular LDGM-LDPC codes over the binary-erasure channel (BEC) under iterative-decoding with message-passing is described.

DOI
10.1109/ciss.2010.5464840

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An irregular LDGM-LDPC code is studied as a sub-code of an LDPC code with some randomly punctured output-bits. It is shown that the LDGM-LDPC codes achieve rates arbitrarily close to the channel-capacity of the binary-input symmetric-output memoryless (BISOM) channel with bounded complexity. The measure of complexity is the average-degree (per information-bit) of the check-nodes for the factor-graph of the code. A lower-bound on the average degree of the check-nodes of the irregular LDGM-LDPC codes is obtained. The bound does not depend on the decoder used at the receiver. The stability condition for decoding the irregular LDGM-LDPC codes over the binary-erasure channel (BEC) under iterative-decoding with message-passing is described.
An irregular LDGM-LDPC code is studied as a sub-code of an LDPC code with some randomly \emph{punctured} output-bits. It is shown that the LDGM-LDPC codes achieve rates arbitrarily close to the channel-capacity of the binary-input symmetric-output memoryless (BISOM) channel with bounded \emph{complexity}. The measure of complexity is the average-degree (per information-bit) of the check-nodes for the factor-graph of the code. A lower-bound on the average degree of the check-nodes of the irregular LDGM-LDPC codes is obtained. The bound does not depend on the decoder used at the receiver. The stability condition for decoding the irregular LDGM-LDPC codes over the binary-erasure channel (BEC) under iterative-decoding with message-passing is described.
Introduction Sequences of irregular low-density parity-check (LDPC) codes that achieve the capacity of the binary erasure channel (BEC) under iterative decoding were first constructed by Luby, et al. in [11]. This was followed by the work of Chung, et al., which provided evidence suggesting that sequences of iteratively decoded LDPC codes can also achieve the channel capacity of the binary-input additive white Gaussian noise (AWGN) channel [3]. Since then, density evolution (DE) [13] has been used to optimize irregular LDPC codes for a variety of memoryless channels (e.g., [6]), and the results suggest, for each channel, that sequences of iteratively decoded LDPC codes can indeed achieve the channel capacity. In fact, the discovery of a channel whose capacity cannot be approached by LDPC codes would be more surprising than a proof that iteratively decoded LDPC codes can achieve the capacity of any binary-input symmetric channel (BISC). The idea of decoding a code transmitted over a c
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.
The asymptotic iterative decoding performances of low-density parity-check (LDPC) codes using min-sum (MS) and sum-product (SP) decoding algorithms on memoryless binary-input output-symmetric (MBIOS) channels are analyzed in this paper. For MS decoding, the analysis is done by upper bounding the bit error probability of the root bit of a tree code by the sequence error probability of a subcode of the tree code assuming the transmission of the all-zero codeword. The result is a recursive upper bound on the bit error probability after each iteration. For SP decoding, we derive a recursively determined lower bound on the bit error probability after each iteration. This recursive lower bound recovers the density evolution equation of LDPC codes on the binary erasure channel (BEC) with inequalities satisfied with equalities. A significant implication of this result is that the performance of LDPC codes under SP decoding on the BEC is an upper bound of the performance on all MBIOS channels with the same uncoded bit error probability. All results hold for the more general multi-edge type LDPC codes.
This paper deals with the irregular binary low-density parity-check (LDPC) codes and two iterative low-complexity decoding algorithms. The first one is the majority error-correcting decoding algorithm, and the second one is iterative erasure-correcting decoding algorithm. The lower bounds on correcting capabilities (the guaranteed corrected error and erasure fraction respectively) of irregular LDPC code under decoding (error and erasure correcting respectively) algorithms with low-complexity were represented. These lower bounds were obtained as a result of analysis of Tanner graph representation of irregular LDPC code. The numerical results, obtained at the end of the paper for proposed lower-bounds achieved similar results for the previously known best lower-bounds for regular LDPC codes and were represented for the first time for the irregular LDPC codes.
We study error bounds for linear programming decoding of regular low-density parity-check (LDPC) codes. For memoryless binary-input output-symmetric channels, we prove bounds on the word error probability that are inverse doubly exponential in the girth of the factor graph. For memoryless binary-input AWGN channel, we prove lower bounds on the threshold for regular LDPC codes whose factor graphs have logarithmic girth under LP-decoding. Specifically, we prove a lower bound of σ = 0.735 (upper bound of [(Eb)/(N0)]=2.67 dB) on the threshold of (3, 6)-regular LDPC codes whose factor graphs have logarithmic girth. Our proof is an extension of a recent paper of Arora, Daskalakis, and Steurer [STOC 2009] who presented a novel probabilistic analysis of LP decoding over a binary symmetric channel. Their analysis is based on the primal LP representation and has an explicit connection to message passing algorithms. We extend this analysis to any MBIOS channel.