On Perfect Sequence Covering Arrays

A PSCA\((v, t, \lambda )\) is a multiset of permutations of the v-element alphabet \(\{0, \dots , v-1\}\), such that every sequence of t distinct elements of the alphabet appears in the specified order in exactly \(\lambda \) of the permutations. For \(v \geqslant t \geqslant 2\), we define g(v, t) to be the smallest positive integer \(\lambda \), such that a PSCA\((v, t, \lambda )\) exists. We show that \(g(6, 3) = g(7, 3) = g(7, 4) = 2\) and \(g(8, 3) = 3\). Using suitable permutation representations of groups, we make improvements to the upper bounds on g(v, t) for many values of \(v \leqslant 32\) and \(3\leqslant t\leqslant 6\). We also prove a number of restrictions on the distribution of symbols among the columns of a PSCA.