The existence of a class of Kirkman squares of index 2

A Kirkman square with index λ, latinicity μ, block size k and ν points, KSk(v; μ, λ), is a t × t array (t = λ(ν−1)/μ(k − 1)) defined on a ν-set V such that (1) each 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 k-subset of V, and (3) the collection of blocks obtained from the nonempty cells of the array is a (ν, k, λ)-BIBD. For μ = 1, the existence of a KSk(ν; μ, λ) is equivalent to the existence of a doubly resolvable (ν, k, λ)-BIBD. In this case the only complete results are for k = 2. The case k = 3, λ = 1 appears to be quite difficult although some existence results are available. For k = 3, λ = 2 the problem seems to be more tractable. In this paper we prove the existence of a KS3(ν; 1, 2) for all ν ≡ 3 (mod 12).