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

Abstract

Let $V$ be a set of $v$ elements. Let $G_1,G_2,\dots ,G_m$ be a partition of $V$ into $m$ sets. A $\{G_1,G_2,\dots ,G_m\}$-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 $i=1,2,\dots ,m$. (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 $GDD(v;k;G_1,G_2,\dots ,G_m;0,\lambda )$. If $|G_i|=h$ for $i=1,2,\dots ,m$, 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 $K{S}_3(v;1,2)s$. 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 $m\equiv 1(\mathrm{mod}12)$. These frames can be used to construct $K{S}_3(v;2,4)s$.