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

A $\mu$-way $(v,k,t)$ trade $T=\{T_1,T_2,\mathrm{.}\mathrm{.}\mathrm{.},T_{\mu }\}$ of volume $m$ consists of $\mu$ disjoint collections $T_{1}, T_{2}, \ldots, T_{\mu}$,eachof$m$blocksofsize$k$,suchthatforevery$t$-subsetof$v$-set$V$thenumberofblockscontainingthis$t$-subsetisthesameineach$T_{i}$(for$1 \leq i \leq \mu$)\mathrm{.}A$\mu$-way$(v, k, t)$tradeiscalled$\mu$-way$(v, k, t)$Steinertradeifany$t$-subsetoffound$(T)$ 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 $d\in {4,5,6}$,exceptforsevensmallvaluesof$v$.