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

Balaji Sundar RAJAN; Ganapathy VISWANATH

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

Balaji Sundar RAJAN, Ganapathy VISWANATH

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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.

Publication: IEICE TRANSACTIONS on Fundamentals Vol.E86-A No.2 pp.480-486

Publication Date: 2003/02/01

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Type of Manuscript: PAPER

Category: Coding Theory

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Balaji Sundar RAJAN, Ganapathy VISWANATH, "On Asymptotic Elias Bound for Euclidean Space Codes over Distance-Uniform Signal Sets" in IEICE TRANSACTIONS on Fundamentals, vol. E86-A, no. 2, pp. 480-486, February 2003, doi: .
Abstract: 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.
URL: https://global.ieice.org/en_transactions/fundamentals/10.1587/e86-a_2_480/_p

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@ARTICLE{e86-a_2_480,
author={Balaji Sundar RAJAN, Ganapathy VISWANATH, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={On Asymptotic Elias Bound for Euclidean Space Codes over Distance-Uniform Signal Sets},
year={2003},
volume={E86-A},
number={2},
pages={480-486},
abstract={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.},
keywords={},
doi={},
ISSN={},
month={February},}

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TY - JOUR
TI - On Asymptotic Elias Bound for Euclidean Space Codes over Distance-Uniform Signal Sets
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 480
EP - 486
AU - Balaji Sundar RAJAN
AU - Ganapathy VISWANATH
PY - 2003
DO -
JO - IEICE TRANSACTIONS on Fundamentals
SN -
VL - E86-A
IS - 2
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - February 2003
AB - 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.
ER -