The existence of doubly resolvable (v, 3, 2)-BIBDs

A Kirkman square with index λ, latinicity μ, block size κ, and ν points, a KSκ(ν; μ, λ), is a t × t array (t = λ(ν − 1)μ(κ − 1)) defined on a ν-set V such that (1) every point of V is contained in precisely μ cells of each row and column, (2) each cell of the array is either empty or contains a κ-subset of V, and (3) the collection of blocks obtained from the non-empty cells of the array is a (ν, κ, λ)-BIBD. For μ = 1, the existence of a KSκ(ν; μ, λ) is equivalent to the existence of a doubly resolvable (ν, κ, λ)-BIBD. The spectrum of KS2(ν; 1, 1) or Room squares was completed by Mullin and Wallis in 1975. In this paper, we determine the spectrum of KS3(ν; 1, 2) or DR(ν, 3, 2)-BIBDs with at present six possible exceptions for ν.