Balanced ternary designs with block size three, any Λ and anyR

Abstract

A balanced ternary design on V elements is a collection of B blocks (which are multisets) of size K, such that each element occurs 0, 1 or 2 times per block and R times altogether, and such that each unordered pair of distinct elements occurs Λ times. (For example, in the block xxyyz, the pair xy is said to occur four times and the pairs xz, yz twice each.) It is straightforward to show that each element has to occur singly in a constant number of blocks, say ρ1, and so each element also occurs twice in a constant number of blocks, say ρ2, where R=ρ1+2 ρ2. If ρ2=0 the design is a balanced incomplete block design (binary design), so we assume ρ2>0, and K 1 if ρ2>0 (and K>2). In 1980 and 1982 the author gave necessary and sufficient conditions for the existence of balanced ternary designs with K=3, Λ=2 and ρ2=1, 2 or 3. In this paper work on the existence of balanced ternary designs with block size three is concluded, in that necessary and sufficient conditions for the existence of a balanced ternary design with K=3, any Λ>1 and any ρ2 are given.