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

Abstract

A $$\mu $$ -way (v, k, t) trade is a pair $$T=(X,\{T_1,T_2,\ldots , T_{\mu }\})$$ such that for eacht-subset of v-set X the number of blocks containing this t-subset is the same in each $$T_i$$ $$(1\le i\le \mu )$$ . In the other words for each $$1\le i<j\le \mu $$ , the pair $$(X,\{T_i,T_j\})$$ is a (v, k, t) trade. A $$\mu $$ -way (v, k, t) trade $$T=(X,\{T_1,T_2,\ldots , T_{\mu }\})$$ with any t-subset occuring at most once in $$T_i$$ $$(1\le i\le \mu )$$ is said to be a $$\mu $$ -way (v, k, t) Steiner trade. The trade is called d-homogeneous if each point occurs in exactly d blocks of $$T_i$$ . In this paper, we construct d-homogeneous $$\mu $$ -way (v, 3, 2) Steiner trades with the first, second and third smallest volume for each $$d\equiv 0$$ (mod 3) and possible $$\mu $$ . Also, we show that for each $$d\equiv 0$$ (mod 3) there exist d-homogeneous $$\mu $$ -way (v, 3, 2) Steiner trades for sufficiently large values of v.