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arXiv:cs/0412111

On the asymptotic accuracy of the union bound

Alexander Barg

Published Dec 24, 2004• 3 citations
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Abstract

A new lower bound on the error probability of maximum likelihood decoding of a binary code on a binary symmetric channel was proved in Barg and McGregor (2004, cs.IT/0407011). It was observed in that paper that this bound leads to a new region of code rates in which the random coding exponent is asymptotically tight, giving a new region in which the reliability of the BSC is known exactly. The present paper explains the relation of these results to the union bound on the error probability.

DOI
10.48550/arxiv.cs/0412111

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A new lower bound on the error probability of maximum likelihood decoding of a binary code on a binary symmetric channel was proved in Barg and McGregor (2004, cs.IT/0407011). It was observed in that paper that this bound leads to a new region of code rates in which the random coding exponent is asymptotically tight, giving a new region in which the reliability of the BSC is known exactly. The present paper explains the relation of these results to the union bound on the error probability.
New lower bounds on the error probability of block codes with maximum-likelihood decoding are proposed. The bounds are obtained by applying a new lower bound on the probability of a union of events, derived by improving on de Caen's lower bound. The new bound includes an arbitrary function to be optimized in order to achieve the tightest results. Since the optimal choice of this function is known, but leads to a trivial and useless identity, we find several useful approximations for it, each resulting in a new lower bound. For the additive white Gaussian noise (AWGN) channel and the binary-symmetric channel (BSC), the optimal choice of the optimization function is stated and several approximations are proposed. When the bounds are further specialized to linear codes, the only knowledge on the code used is its weight enumeration. The results are shown to be tighter than the latest bounds in the current literature, such as those by Seguin (1998) and by Keren and Litsyn (2001). Moreover, for the BSC, the new bounds widen the range of rates for which the union bound analysis applies, thus improving on the bound to the error exponent compared with the de Caen-based bounds.
Bounds on the error probability of maximum likelihood decoding of a binary linear code are considered. The bounds derived use the weight spectrum of the code and they are tighter than the conventional union bound in the case of large noise in the channel. The bounds derived are applied to a code with an average spectrum, and the result is compared to the random coding exponent. The author shows that the bound considered for the binary symmetrical channel case coincides asymptotically with the random coding bound. For the case of AWGN channel the author shows that Berlekamp's (1980) tangential bound can be improved, but even this improved bound does not coincide with the random coding bound, although it can be very close to it.>
We derive both upper and lower bounds on the decoding error probability of maximum-likelihood (ML) decoded low-density parity-check (LDPC) codes. The results hold for any binary-input symmetric-output channel. Our results indicate that for various appropriately chosen ensembles of LDPC codes, reliable communication is possible up to channel capacity. However, the ensemble averaged decoding error probability decreases polynomially, and not exponentially. The lower and upper bounds coincide asymptotically, thus showing the tightness of the bounds. However, for ensembles with suitably chosen parameters, the error probability of almost all codes is exponentially decreasing, with an error exponent that can be set arbitrarily close to the standard random coding exponent.
We derive new upper bounds on the error exponents for the maximum-likelihood decoding and error detecting in the binary symmetric channels. This is an improvement on the best earlier known bounds by Shannon-Gallager-Berlekamp (1967) and McEliece-Omura (1977). For the probability of undetected error the new bounds are better than the bounds by Levenshtein (1978, 1989) and the bound by Abdel-Ghaffar (see ibid., vol.43, p.1489-502, 1997). Moreover, we further extend the range of rates where the undetected error exponent is known to be exact. The new bounds are based on an analysis of possible distance distributions of the codes along with some inequalities relating the distance distributions to the error probabilities.
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.