# Balanced ternary designs with block size three, any Λ and any R

Billington E.J. (1985) Balanced ternary designs with block size three, any Λ and any R. Aequationes Mathematicae, 29 1: 244-289. doi:10.1007/BF02189832

Author Billington E.J. Balanced ternary designs with block size three, any Λ and any R Aequationes Mathematicae   Check publisher's open access policy 0001-9054 1985-01-01 Article (original research) 10.1007/BF02189832 29 1 244 289 46 Birkhäuser-Verlag 2600 Mathematics 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<2 V (corresponding to incompleteness in the binary case). Necessarily Λ>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. AMS (1980) subject classification: Primary 05B05, Secondary 62K10 C1 Provisional Code Unknown

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