On the existence of $d$-homogeneous $3$-way Steiner trades

Abstract

A $\mu$-way $(v,k,t)$ trade $T=\{T_1,T_2,...,T_{\mu }\}$ of volume $m$ consists of $\mu$ disjoint collections $T_1,T_2,\dots ,T_{\mu }$, each of $m$ blocks of size $k$, such that for every $t$-subset of $v$-set $V$ the number of blocks containing this $t$-subset is the same in each $T_i$ (for $1\le i\le \mu$). A $\mu$-way $(v,k,t)$ trade is called $\mu$-way $(v,k,t)$ Steiner trade if any $t$-subset of found$(T)$ occurs at most once in $T_1$ $(T_j,j\ge 2)$. A $\mu$-way $(v,k,t)$ trade is called $d$-homogeneous if each element of $V$ occurs in precisely $d$ blocks of $T_1$ $(T_j,j\ge 2)$. In this paper we characterize the $3$-way $3$-homogeneous $(v,3,2)$ Steiner trades of volume $v$. Also we show how to construct a $3$-way $d$-homogeneous $(v,3,2)$ Steiner trade for $d\in \{4,5,6\}$, except for seven small values of $v$.

11 pages. Utilitas Math.(to appear)

Cited by in corpus (1)

• The minimum possible volume size of u-way (v, k, t) trades