A Completion of the Spectrum of 3-Way $(v,k,2)$ Steiner Trades

A 3-way \((v,k,t)\) trade \(T\) of volume \(m\) consists three pairwise disjoint collections \(T_{1}\), \(T_{2}\) and \(T_{3}\), each blocks size \(k\), such that for every \(t\)-subset \(v\)-set \(V\), the number containing this is same in \(T_{i}\) \(1\leq i\leq 3\). If any found(\(T\)) occurs at most once 3\), then called Steiner trade. We attempt to complete spectrum \(S_{3s}(v,k)\), set all possible sizes, \((v,k,2)\) trades, by applying some block designs, as BIBDs, RBs, GDDs, RGDDs, \(r\times s\) packing grid blocks. Previously, we obtained results about existence trades. In particular, proved there exists a when \(12(k-1)\leq m\) \(15\leq k\) (Rashidi Soltankhah Discrete Math. 339(12): 2955–2963, 2016). Now, show claim correct also \(k\leq 14\).