A group-based structure for perfect sequence covering arrays

Abstract

An (n, k)-perfect sequence covering array with multiplicity $$\lambda $$ , denoted $$\mathrm{{PSCA}}(n,k,\lambda )$$ , is a multiset whose elements are permutations of the sequence $$(1,2, \dots , n)$$ and which collectively contain each ordered length k subsequence exactly $$\lambda $$ times. The primary objective is to determine for each pair (n, k) the smallest value of $$\lambda $$ , denoted g(n, k), for which a $$\mathrm{{PSCA}}(n,k,\lambda )$$ exists; and more generally, the complete set of values $$\lambda $$ for which a $$\mathrm{{PSCA}}(n,k,\lambda )$$ exists. Yuster recently determined the first known value of g(n, k) greater than 1, namely $$g(5,3)=2$$ , and suggested that finding other such values would be challenging. We show that $$g(6,3)=g(7,3)=2$$ , using a recursive search method inspired by an old algorithm due to Mathon. We then impose a group-based structure on a perfect sequence covering array by restricting it to be a union of distinct cosets of a prescribed nontrivial subgroup of the symmetric group $$S_n$$ . This allows us to determine the new results that $$g(7,4)=2$$ and $$g(7,5) \in \{2,3,4\}$$ and $$g(8,3) \in \{2,3\}$$ and $$g(9,3) \in \{2,3,4\}$$ . We also show that, for each $$(n,k) \in \{ (5,3), (6,3), (7,3), (7,4) \}$$ , there exists a $$\mathrm{{PSCA}}(n,k,\lambda )$$ if and only if $$\lambda \ge 2$$ ; and that there exists a $$\mathrm{{PSCA}}(8,3,\lambda )$$ if and only if $$\lambda \ge g(8,3)$$ .