On a pair of dual subschemes of the Hamming scheme \(H_ n(q)\)
type: Item
instance of: scholarly article
title: On a pair of dual subschemes of the Hamming scheme \(H_ n(q)\)
zbMATH Open document ID: 0579.05021
DOI: 10.1016/S0195-6698(85)80004-4
author: Jean-Marie Goethals
author: Robert Calderbank
published in: European Journal of Combinatorics
publication date: 1985-01-01T00:00:00Z
review text: An association scheme with n classes an a set X is a partition of the set of 2-element subsets of X into n classes \(\Gamma_ 1,\Gamma_ 2,...,\Gamma_ n\) satisfying (1) given \(x\in X\), the number \(v_ i\) of \(y\in X\) with \(\{x,y\}\in \Gamma_ i\) depends only on i; (2) given x,y\(\in X\) with \(\{x,y\}\in \Gamma_ k\), the number of \(z\in X\) with \(\{x,z\}\in \Gamma_ i\) and \(\{y,z\}\in \Gamma_ j\) is a constant \(P^ k_{i,j}\) depending only on i,j,k. The Hamming scheme \(H_ n(q)\) is an association scheme with n classes, where \(X=F^ n\), \(F=GF(q)\) and a pair of vectors \(\{\) x,y\(\}\) is in \(\Gamma_ i\) if and only if the Hamming distance \(d(x,y)=i.\) The authors analyze codes in the Hamming association scheme \(H_ n(q)\). The main result is Theorem 1: Let C be a subset of \(H_ n(q)\) with degree \(e+1\) and strength 2e or \(2e+1\). If \(w_ 1,w_ 2,...,w_{e+1}\) are nonzero distances between elements of C then \[ q^{e(e+1)/2}\prod_{i>j}(w_ i-w_ j)/\prod^{e}_{k=1}(k!) \] is an integer dividing \(| C|^{e+1}\). Further, the authors give necessary conditions for the existence of nonlinear uniformly packed codes, and they determine the possible parameters of uniformly packed 2- error-correcting codes.
reviewed by: K.Burian
Mathematics Subject Classification ID: 05B30
Mathematics Subject Classification ID: 94B05
zbMATH DE Number: 3926939
zbMATH Keywords: association scheme
zbMATH Keywords: Hamming scheme
zbMATH Keywords: codes
MaRDI profile type: Publication
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full work available at URL: s0195-6698(85)80004-4
OpenAlex ID: W1988802300
description: scientific article; zbMATH DE number 3926939