The Number of Repeated Blocks in Balanced Ternary Designs with Block Size Three

Abstract

Let $D$ denote any balanced ternary design with block size three, index two, and ${\rho }_2=1$ (that is, with each element occurring repeated in just one block). This paper shows that there exists such a design $D$ on $V$ elements containing exactly $k$ pairs of repeated blocks if and only if $V\equiv 0(\mathrm{mod}3)$, $$0\le k\le t_V=\frac{1}{6}V(V-3),k\ne t_V–1,\text{and}(k,V)\ne (1,6)$$.