On the Existence of \((2,4;3,m,h)\)-Frames for \(h=1,3\) and \(6\)

Let \(V\) be a set of \(v\) elements. Let \(G_1, G_2, \ldots, G_m\) be a partition of \(V\) into \(m\) sets. A \(\{G_1, G_2, \ldots, G_m\}\)-frame \(F\) with block size \(k\), index \(\lambda\) and latinicity \(\mu\) is a square array of side \(v\) which satisfies the properties listed below. We index the rows and columns of \(F\) with the elements of \(V\). (1) Each cell is either empty or contains a \(k\)-subset of \(V\). (2) Let \(F_i\) be the subsquare of \(F\) indexed by the elements of \(G_i\). \(F_i\) is empty for \(i = 1, 2, \ldots, m\). (3) Let \(j \in G_i\). Row \(j\) of \(F\) contains each element of \(V – G_i\) \(\mu\) times and column \(j\) of \(F\) contains each element of \(V – G_i\) \(\mu\) times. (4) The collection of blocks obtained from the nonempty cells of \(F\) is a \(GDD(v; k; G_1, G_2, \ldots, G_m; 0, \lambda)\). If \(|G_i| = h\) for \(i = 1, 2, \ldots, m\), we call \(F\) a \((\mu, \lambda, k, m, h)\)-frame.
Frames with \(\mu=\lambda=1\) and \(k = 2\) were used by D.R. Stinson to establish the existence of skew Room squares and Howell designs. \((1, 2; 3, m, h)\)-frames with \(h = 1, 3\) and \(6\) have been studied and can be used to produce \(KS_3(v; 1, 2)s\). In this paper, we prove the existence of \((2, 4; 3, m, h)\)-frames for \(h = 3\) and \(6\) with a finite number of possible exceptions. We also show the existence of \((2, 4; 3, m, 1)\)-frames for \(m \equiv 1 \pmod{12}\). These frames can be used to construct \(KS_3(v; 2, 4)s\).