On the Existence of d-Homogeneous μ-Way (v, 3, 2) Steiner Trades.

A μ-way (v, k, t) trade is a pair T=(X,{T1,T2,...,Tμ}) such that for eacht-subset of v-set X the number of blocks containing this t-subset is the same in each Ti(1≤i≤μ). In the other words for each 1≤i<j≤μ, the pair (X,{Ti,Tj}) is a (v, k, t) trade. A μ-way (v, k, t) trade T=(X,{T1,T2,...,Tμ}) with any t-subset occuring at most once in Ti(1≤i≤μ) is said to be a μ-way (v, k, t) Steiner trade. The trade is called d-homogeneous if each point occurs in exactly d blocks of Ti. In this paper, we construct d-homogeneous μ-way (v, 3, 2) Steiner trades with the first, second and third smallest volume for each d≡0 (mod 3) and possible μ. Also, we show that for each d≡0 (mod 3) there exist d-homogeneous μ-way (v, 3, 2) Steiner trades for sufficiently large values of v. [ABSTRACT FROM AUTHOR]