A Completion of the spectrum of 3-way $(v,k,2)$ Steiner trades

Abstract

A 3-way $(v,k,t)$ trade $T$ of volume $m$ consists of three pairwise disjoint collections $T_1$, $T_2$ and $T_3$, 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 $1\leq i\leq 3$. If any $t$-subset of found($T$) occurs at most once in each $T_i$ for $1\leq i\leq 3$, then $T$ is called 3-way $(v,k,t)$ Steiner trade. We attempt to complete the spectrum $S_{3s}(v,k)$, the set of all possible volume sizes, for 3-way $(v,k,2)$ Steiner trades, by applying some block designs, such as BIBDs, RBs, GDDs, RGDDs, and $r\times s$ packing grid blocks. Previously, we obtained some results about the existence some 3-way $(v,k,2)$ Steiner trades. In particular, we proved that there exists a 3-way $(v,k,2)$ Steiner trade of volume $m$ when $12(k-1)\leq m$ for $15\leq k$ (Rashidi and Soltankhah, 2016). Now, we show that the claim is correct also for $k\leq 14$.