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IEEE Information Theory Workshop (ITW)

A coset weight count that proves that the simplex codes are not optimal for error correction

Tor Helleseth, Torleiv Kløve, V.I. Levenshtein

Published Mar 1, 2004• 2 citations
Publisher

Abstract

The number of cosets of weight 2/sup k-2/ or less are determined for the [2/sup k/-1, k, 2/sup k-1/] simplex code and a [2/sup k/-1, k, 2/sup k-1/-1] code obtained by a simple modification of the simplex code. The result proves that the [2/sup k/-1, k] simplex codes are not optimal for error correction on the binary symmetric channel with small bit error probability, p, (for k/spl ges/3). A proof that the modified code is better for all p, 0<p<1/2, is sketched.

DOI
10.1109/itw.2003.1216738

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The number of cosets of weight 2/sup k-2/ or less are determined for the [2/sup k/-1, k, 2/sup k-1/] simplex code and a [2/sup k/-1, k, 2/sup k-1/-1] code obtained by a simple modification of the simplex code. The result proves that the [2/sup k/-1, k] simplex codes are not optimal for error correction on the binary symmetric channel with small bit error probability, p, (for k/spl ges/3). A proof that the modified code is better for all p, 0<p<1/2, is sketched.
This paper focuses on error thresholds for Pauli channels. We numerically compute lower bounds for the thresholds using the analytic framework of coset weight enumerators pioneered by DiVincenzo, Shor and Smolin in 1998. In particular, we study potential non-additivity of a variety of small stabilizer codes and their concatenations, and report several new concatenated stabilizer codes of small length that show significant non-additivity. We also give a closed form expression of coset weight enumerators of concatenated phase and bit flip repetition codes. Using insights from this formalism, we estimate the threshold for concatenated repetition codes of large lengths. Finally, for several concatenations of small stabilizer codes we optimize for channels which lead to maximal non-additivity at the hashing point of the corresponding channel. We supplement these results with a discussion on the performance of various stabilizer codes from the perspective of the non-additivity and threshold problem. We report both positive and negative results, and highlight some counterintuitive observations, to support subsequent work on lower bounds for error thresholds.
Complete decoding of systematic linear block codes by standard arrays with coset leaders not necessarily of minimal weight gives, in several interesting cases on the binary symmetric channel with error probabilityp, a smaller symbol error probability than decoding using the coset leaders of minimum weight. The proposed choice of the coset leaders allows for the classes of uniformly packed and nearly perfect binary codes to achieve a lower information bit error probability, at least forpsufficiently near zero.
The well-known m-out-of-n code /spl Omega//sub n//sup m/ consists of all binary vectors of length n and weight m. It is known that it is good for error detection (in the technical sense, that is, the probability of undetected error P/sub ud/(/spl Omega//sub n//sup m/,p)/spl les/P/sub ud/(/spl Omega//sub n//sup m/,1/2) for all p, 0/spl les/p/spl les/1/2) only for a few small values of m and n. It is therefore of interest to determine (bounds for) the threshold in general, that is, find the range of bit-error probabilities p for which P/sub ud/ (/spl Omega//sub n//sup m/,p)/spl les/P/sub ud/ (/spl Omega//sub n//sup m/,1/2). In this article such bounds are given.