PBIB designs and association schemes obtained from finite rings (Q1613506)

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.