On Asymptotic Elias Bound for Euclidean Space Codes over Distance-Uniform Signal Sets

Balaji Sundar RAJAN

IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E86-A    No.2    pp.480-486
Publication Date: 2003/02/01
Online ISSN: 
Print ISSN: 0916-8508
Type of Manuscript: PAPER
Category: Coding Theory
Euclidean space codes,  group codes,  uniform signal sets,  signal sets matched to groups,  

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The asymptotic Elias upper bound of codes designed for Hamming distance is well known. Piret and Ericsson have extended this bound for codes over symmetric PSK signal sets with Euclidean distance and for codes over signal sets that form a group, with general distance function respectively. The tightness of these bounds depend on a choice of a probability distribution, and finding the distribution (optimum distribution) that leads to the tightest bound is difficult in general. In this paper we point out that these bounds are valid for codes over the wider class of distance-uniform signal sets (a signal set is referred to be distance-uniform if the Euclidean distance distribution is same from any point of the signal set). We show that optimum distributions can be found for (i) simplex signal sets, (ii) Hamming spaces and (iii) biorthogonal signal set. The classical Elias bound for arbitrary alphabet size is shown to be obtainable by specializing the extended bound to simplex signal sets with optimum distribution. We also verify Piret's conjecture for codes over 5-PSK signal set.