On the Existence of d-Homogeneous $\mu$-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.