More balanced ternary designs with block size three

Abstract

A balanced ternary design (BTD) is a balanced design on V elements with constant block size, in which each element may occur 0, 1 or 2 times in each block. Thus blocks may be collections of elements rather than subsets of elements. A regular BTD is one in which each element occurs singly in ϱ 1 blocks, and is repeated in ϱ 2 blocks. The condition V ≡ 0( mod 3) was shown in an earlier paper to be both necessary and sufficient for the existence of a regular BTD on V elements with block size 3, index 2 and ϱ 2=1. We show in this paper that if ϱ 2 = 2, necessary and sufficient conditions for existence of a regular BTD with block size 3 and index 2 are V ≡0 or 2( mod 3), V ≥5, and if ϱ 2=3 the necesary and sufficient conditions are V ≡ 0 or 1( mod 3), V ≥ 7. Hence by allowing ϱ 2 to be as large as 3, we have a regular BTD on V elements with block size 3 and index 2 for all V ≠1,2,4.