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 in Discrete Math. 339(12): 2955–2963, 2016). Now, we show that the claim is correct also for \(k\leq 14\).