Constant weight codes and group divisible designs
type: Item
instance of: scholarly article
title: Constant weight codes and group divisible designs
zbMATH Open document ID: 0945.05011
DOI: 10.1023/A:1008318207622
author: Simon Blake-Wilson
author: Kevin T. Phelps
published in: Designs, Codes and Cryptography
publication date: 2000-10-08T00:00:00Z
review text: This paper continues the study of optimal constant weight codes over arbitrary alphabets which was initiated by \textit{T. Etzion} [Optimal constant weight codes over \(Z_k\) and generalized designs, Discrete Math. 169, No. 1-3, 55-82 (1997)]. Etzion showed that such codes are equivalent to special GDD's known as generalized Steiner systems \(\text{GS}(t, k,n,g)\). This paper presents recursive constructions for these designs when \(t=2\) and \(k=3\). These constructions allow the authors to prove that for a fixed group size \(g\), the spectrum of \(\text{GS}(2,3,n,g)\) is PBD-closed. This enables them to show that the necessary numerical conditions on the values of \(n\) are asymptotically sufficient. Since this paper was published subsequent work has concentrated on finding the exact spectrum of \(n\) for which there exists a \(\text{GS}(2,3,n,g)\) for a fixed small group size \(g\).
Mathematics Subject Classification ID: 05B07
Mathematics Subject Classification ID: 94B25
Mathematics Subject Classification ID: 05B05
Mathematics Subject Classification ID: 94B60
Mathematics Subject Classification ID: 05B15
zbMATH DE Number: 1275100
zbMATH Keywords: optimal constant weight codes
zbMATH Keywords: generalized Steiner systems
reviewed by: J. H. Dinitz
MaRDI profile type: Publication
full work available at URL: a:1008318207622
OpenAlex ID: W1500003238
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description: scientific article; zbMATH DE number 1275100