PBIB designs and association schemes obtained from finite rings

We assume (R,+,·) is a ring with unit. Let U(R) denote the set of invertible elements and suppose Φ is a subgroup of U(R) with −1∈Φ. Define an equivalence relation ∼ on R∗=R⧹{0} by s1∼s2 if there is b∈Φ such that bs1=s2. Let s1,s2,…,sm be representatives of the distinct equivalence classes. Define Ai={{x,y} | (y−x)∼si} for i=1,2,…,m. We prove that (R,A) is an association scheme, where A={Ai | i=1,2,…,m}. Next suppose R is finite and let S be a proper subset of R with |S|⩾2. Define B={bS+a | b∈Φ, a∈R}. Then (R,B,A) is a partially balanced incomplete block design (PBIBD). Moreover, if S satisfies S≠−S+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.