More balanced ternary designs with block size three

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{variant} blocks, and is repeated in ρ{variant} 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{variant}=1. We show in this paper that if ρ{variant} = 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{variant}=3 the necesary and sufficient conditions are V ≡ 0 or 1(mod 3), V ≥ 7. Hence by allowing ρ{variant} 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