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IEEE Transactions on Information Theory

LP Decoding of Regular LDPC Codes in Memoryless Channels

Nissim Halabi, Guy Even

Published Jan 21, 2011• 22 citations
Publisher

Abstract

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.

DOI
10.1109/tit.2010.2094830

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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.
We study error bounds for linear programming decoding of regular 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=0.735σ=0.735σ=0.735 (upper bound of EbN0=2.67\frac{Eb}{N_0}=2.67N0​Eb​=2.67dB) on the threshold of (3,6)(3,6)(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.
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 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.
A design of robust error-correcting codes that achieve reli able communication over various channels is of great theoretical and practical interest. Such codes are termed universal. This paper considers the universality of lowdensity parity-check (LDPC) code ensembles over families of memoryless binary-input output-symmetric (MBIOS) channels. Universality is considered both under belief-pr opagation (BP) and maximum-likelihood (ML) decoding. For the BP decoding case, we derive a density-evolution-based analytical method for designing LDPC code ensembles that are universal over various families of MBIOS channels. We also derive a necessary condition for universality of LDPC code ensembles under BP decoding; this condition is used to provide bounds on the universally achievable fraction of capacity. These results enable us to provide conditions for reliable / unreliable communications under BP decoding that are based on the Bhattacharyya parameter of the channel. For the ML decoding case, we prove that properly selected regular LDPC code ensembles are universally capacity-achieving for the set of equicapacity MBIOS channels and extend this result to punctured regular LDPC code ensembles. Index Terms Belief propagation (BP), Bhattacharyya parameter (B-parameter), density evolution (DE), linear programming (LP) bounds, low-density parity-check (LDPC) codes, maximum-likelihood (ML), memoryless binary-input outputsymmetric (MBIOS) channels, stability.
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.