On a conjecture of Mahmoodian and Soltankhah regarding the existence of \((v,k,t)\) trades (Q2715951)

Write \(T = T_1 - T_2\) to denote two disjoint collections \(T_1\) and \(T_2\) of \(k\)-subsets (blocks) of a \(v\)-element set \(V\) (the inequality \(v \leq |V|\) in the paper seems to be a typo). Call it a \((v,k,t)\)-trade of volume \(s\), if each \(t\)-subset of \(V\) is contained in the same number of blocks from \(T_1\) and \(T_2\), and if both \(T_1\) and \(T_2\) consist of \(s\) blocks. Then \(s \geq 2^t\). All trades with \(s = 2^t\) (the so-called basic trades) have been described by \textit{H. L. Hwang} [J. Stat. Plann. Inference 13, 179-191 (1986; Zbl 0593.62071)]. Using a sum of two basic trades, the authors prove that for every \(i\), \(0 \leq i \leq t\), there exists a \((v,k,t)\)-trade of volume \(2^{t+1}-2^{t-i}\). This makes it possible to solve positively the existence problem for any \(s \geq (2t-1)2^t\).