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IEEE International Symposium on Information Theory (ISIT)

Lower bounds on the spectrum and error rate LDPC code ensembles

Ohad Barak, David Burshtein

Published Jan 1, 2005• 8 citations
Publisher

Abstract

We consider the ensemble of regular LDPC codes and obtain an expression for the second moment of the distance spectrum. We show how this expression can be used to derive a lower bound on the probability that the growth rate of a randomly chosen code from the ensemble is equal to the growth rate of the average distance spectrum, when the block length is sufficiently large. In particular, when the connectivity of the code is sufficiently large, the distance spectrum of a code in the ensemble is concentrated. We then derive a lower bound on the probability (confidence level) that the minimum distance and error rate, respectively, of a randomly chosen code from the ensemble are upper and lower bounded by some values (which depend on the confidence level)

DOI
10.1109/isit.2005.1523289

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An ensemble of(J,K)(J,K)(J,K)-regular low-density parity- check (LDPC) convolutional codes is introduced and existence-type lower bounds on the minimum distancedLd _ {\rm L}dL​of code segments of finite lengthLLLand on the free distancedfreed _{\rm free}dfree​are derived. For sufficiently large constraint lengthsν\nuν, the distances are shown to grow linearly withν\nuνand the ratiodL/νd_ {\rm L}/\nudL​/νapproaches the ratiodfree/νd _{ {\rm free}}/\nudfree​/νfor largeLLL. Moreover, the ratio of free distance to constraint length is several times larger than the ratio of minimum distance to block length for Gallager's ensemble of (J,K)-regular LDPC block 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.
In this paper, we derive an upper bound for the average block error probability of a standard irregular low-density parity-check (LDPC) code ensemble under the maximum-likelihood (ML) decoding. Moreover, we show that the upper bound asymptotically decreases polynomially with the code length. Furthermore, when we consider several regular LDPC code ensembles as special cases of standard irregular ones over an additive white Gaussian noise channel, we numerically show that the signal-to-noise ratio (SNR) thresholds at which the proposed bound converges to zero as the code length tends to infinity are smaller than those for a bound provided by Miller et al. We also give an example of a standard irregular LDPC code ensemble which has a lower SNR threshold than a given regular LDPC code ensemble.
We derive bounds on the error probability of ML decoded LDPC (low density parity check) codes, for any binary-input symmetric-output channel. For appropriately chosen ensembles of LDPC codes, reliable communication is possible up to channel capacity. The lower and upper bounds coincide asymptotically, indicating a polynomially decreasing ensemble averaged error probability. For ensembles with suitably chosen parameters, the error probability of almost all codes is exponentially decreasing. Furthermore, the error exponent can be set arbitrarily close to the standard random coding exponent.