In 1992, Mahmoodian and Soltankhah conjectured that, for all \(0 \leq i \leq t\), a \((v, k, t)\) trade of volume \(2^{t+1} – 2^{t-i}\) exists. In this paper we prove this conjecture and, as a corollary, show that if \(s \geq (2t – 1)2^t\) then there exists a \((v, k, t)\) trade of volume \(s\).