Resolvable perfect cyclic designs

Abstract

Let k, λ, and υ be positive integers. A perfect cyclic design in the class PD(υ, k, λ) consists of a pair (Q, B) where Q is a set with |Q| = υ and B is a collection of cyclically ordered k-subsets of Q such that every ordered pair of elements of Q are t apart in exactly λ of the blocks for t = 1, 2, 3,…, k−1. To clarify matters the block [a₁, a₂, …, a_k] has cyclic order a₁ < a₂ < a₃ … < a_k < a₁ and a_i and a_i+1 are said to be t apart in the block where i + t is taken mod k. In this paper we are interested only in the cases where λ = 1 and υ ≡ 1 mod k. Such a design has $\text{υ(υ − 1)}\text{k}$ blocks. If the blocks can be partitioned into υ sets containing $\text{(υ − 1)}\text{k}$ pairwise disjoint blocks the design is said to be resolvable, and any such partitioning of the blocks is said to be a resolution. Any set of $\text{υ − 1)}\text{k}$ pairwise disjoint blocks together with a singleton consisting of the only element not in one of the blocks is called a parallel class. Any resolution of a design yields υ parallel classes. We denote by RPD(υ, k, 1) the class of all resolvable perfect cyclic designs with parameters υ, k, and 1. Associated with any resolvable perfect cyclic design is an orthogonal array with k + 1 columns and υ rows with an interesting conjugacy property. Also a design in the class RPD(υ, k, 1) is constructed for all sufficiently large υ with υ ≡ 1 mod k.