PBIB designs and association schemes obtained from finite rings
type: Item
instance of: scholarly article
title: PBIB designs and association schemes obtained from finite rings
zbMATH Open document ID: 0993.05023
DOI: 10.1016/S0012-365X(01)00231-X
author: Hsin-Min Sun
published in: Discrete Mathematics
publication date: 2002-08-29T00:00:00Z
review text: In a ring \(R\) with \(1\), let \(S\) be a subgroup of the multiplicative group of invertible elements \(U(R)\), containing \(-1\). Define an equivalence relation \(\sim\) on \(R^* = R\setminus \{0\}\) by \(s\sim t\) if there exists \(b\) in \(S\) such that \(bs = t.\) Let \(\{s_1, s_2,\ldots, s_m\}\) be a set of representatives of distinct equivalence classes and define the family of sets \(A_i = \{\{x,y\} \mid (y-x)\sim s_i\},\) for \( i = 1,2,\ldots ,m.\) The author proves that \(R\) together with this family of sets is an association scheme. If \(R\) is finite, \(T\) is a proper subset of \(R\) with card \(T\) greater than or equal to \(2\) and \(B\) is the set of subsets of \(R B = \{ bT + a \mid b \in S\), \(a \in R \}\), then the above scheme together with \(B\) is a PBIBD. Moreover, if \(T\) is not of the form \((-T + a)\) for any \(a\) , then the above PBIBD can be partitioned into two isomorphic PBIBDs. The application of PBIBDs to constant weight codes is introduced.
reviewed by: Mirela Ştefănescu
Mathematics Subject Classification ID: 05B05
Mathematics Subject Classification ID: 05E30
Mathematics Subject Classification ID: 94B25
zbMATH DE Number: 1792432
zbMATH Keywords: association scheme
zbMATH Keywords: partially balanced incomplete block design
zbMATH Keywords: PBIBD
zbMATH Keywords: ring
MaRDI profile type: Publication
Recommended article: Q5586419
Recommended article: Q3803128
Recommended article: PBIB Designs and Their Association Schemes Arising from Minimum Bi-Independent Dominating Sets of Circulant Graphs
Recommended article: Some constructions of PBIB designs
Recommended article: Construction and Combinatorial Problems of Higher Associate Cyclic Pbib Designs
Recommended article: Q4316417
Recommended article: Q5525130
Recommended article: PBIB-designs and association schemes arising from minimum bi-connected dominating sets of hypercubes
description: scientific article; zbMATH DE number 1792432