Gray and Ramsay [5] showed that for any \(s \geq (2t – 1)2^t\), a \(t-(v,k)\) trade of volume \(s\) exists. In this note we improve their bound and show that for \(t \geq 3\), a given \(k\), and \(s \geq (t – 2)2^t + 2^{t-1} + 2\), there exists a simple \(t-(v,k)\) trade of volume \(s\).