A new construction for µ-way Steiner trades

A \mu -way (v,k,t) trade T of volume m consists of \mu pairwise disjoint collections T1... ,Tμ , each of m blocks of size k such that for every t -subset of a v -set V, the number of blocks containing this t -subset is the same in each T_i for 1≤ i≤ \mu . If any t -subset of the v -set V occurs at most once in each T_i for 1≤ i≤ \mu , then T is called a \mu -way (v,k,t) Steiner trade. In 2016, it was proved that there exists a 3-way (v,k,2) Steiner trade of volume m when 12(k-1)≤ m for each k . Here we improve the lower bound to 8(k-1) for even k , by  using a recursive construction.