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IEEE Transactions on Information Theory

On the Error Exponents of Improved Tangential Sphere Bounds

Moshe Twitto, Igal Sason

Published Mar 1, 2007• 14 citations
Publisher

Abstract

The performance of maximum-likelihood (ML) decoded binary linear block codes over the additive white Gaussian noise (AWGN) channel is addressed via the tangential sphere bound (TSB) and two of its recent improved versions. The correspondence is focused on the derivation of the error exponents of these bounds. Although it was shown that some recent improvements of the TSB tighten this bound for finite-length codes, it is demonstrated in this correspondence that their error exponents coincide. For an arbitrary ensemble of binary linear block codes, the common value of these error exponents is explicitly expressed in terms of the asymptotic growth rate of the average distance spectrum

DOI
10.1109/tit.2006.890725

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The performance of maximum-likelihood (ML) decoded binary linear block codes over the additive white Gaussian noise (AWGN) channel is addressed via the tangential sphere bound (TSB) and two of its recent improved versions. The correspondence is focused on the derivation of the error exponents of these bounds. Although it was shown that some recent improvements of the TSB tighten this bound for finite-length codes, it is demonstrated in this correspondence that their error exponents coincide. For an arbitrary ensemble of binary linear block codes, the common value of these error exponents is explicitly expressed in terms of the asymptotic growth rate of the average distance spectrum
The performance of maximum-likelihood (ML) decoded binary linear block codes over the AWGN channel is addressed via the tangential-sphere bound (TSB) and two of its recent improved versions. The paper is focused on the derivation of the error exponents of these bounds. Although it was exemplified that some recent improvements of the TSB tighten this bound for finite-length codes, it is demonstrated in this paper that their error exponents coincide. For an arbitrary ensemble of binary linear block codes, the common value of these error exponents is explicitly expressed in terms of the asymptotic growth rate of the average distance spectrum.
The performance of maximum-likelihood (ML) decoded binary linear block codes over the AWGN channel is addressed via the tangential-sphere bound (TSB) and two of its recent improved versions. Although it was exemplified that some recent improvements of the TSB tighten this bound for finite-length codes, it is demonstrated in this paper that their error exponents coincide. For an arbitrary ensemble of binary linear block codes, the common value of these error exponents is explicitly expressed in terms of the asymptotic growth rate of the average distance spectrum.
Performance evaluation of maximum-likelihood (ML) soft-decision-decoded binary block codes is usually carried out using bounding techniques. Many tight upper bounds on the error probability of binary codes are based on the so-called Gallager's first bounding technique (GFBT). The tangential sphere bound (TSB) of Poltyrev which has been believed for many years to offer the tightest bound developed for binary block codes is an example. Within the framework of the TSB and GFBT, we apply a new method referred to as the "added-hyper-plane" (AHP) technique, to the decomposition of the error probability. This results in a bound developed upon the application of two stages of the GFBT with two different Gallager regions culminating in a tightened upper bound beyond the TSB. The proposed bound is simple and only requires the spectrum of the binary code.
An efficient numerical approach to the exact ensemble distance spectrum of Gallager codes has been developed by evaluating powers of polynomials. With the exact ensemble distance spectrum of Gallager codes, tangential sphere upper bounds on their maximum likelihood (ML) decoding performance over binary input AWGN channels are investigated. Numerical results indicate improved bounds have been obtained, better than Sason and Shamai's results (which are based on Gallager's upper bound on the ensemble distance spectrum), especially in the error floor region. Furthermore, some critical properties of Gallager codes, including typical minimum distance and the performance tradeoff in the waterfall and error floor regions, have been considered.
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.>
This paper presents a method to calculate the exact average block error probability of some random code ensembles under maximum-likelihood decoding. The proposed method is applicable to various channels and ensembles. The focus is on both spherical and Gaussian random codes on the additive white Gaussian noise channel as well as binary random codes on both the binary symmetric channel and the binary erasure channel. While for the uniform spherical ensemble Shannon, in 1959, argued with solid angles in NNN-dimensional space, the presented approach projects the problem into two dimensions and applies standard trigonometry. This simplifies the derivation and also allows for the analysis of the independent identically distributed (i.i.d.) Gaussian ensemble which turns out to perform better for short blocklengths and high rates. Moreover, a new lower bound on the average block error probability of the uniform spherical ensemble is found. For codes with more than three codewords, it is tighter than the sphere packing bound, but requires exactly the same computing effort. Furthermore, tight approximations are proposed to simplify the computation of both the exact average error probability and the two bounds. For the binary symmetric channel and the binary erasure channel, bounds on the average block error probability for i.i.d.\ random coding are derived and compared to the exact calculations.