A polynomial construction of perfect sequence covering arrays

Abstract

A PSCA(v,t,λ) is a multiset of permutations of the v-element alphabet {0,⋯,v-1} such that every sequence of t distinct elements of the alphabet appears in the specified order in exactly λ permutations. For v⩾t, let g(v,t) be the smallest positive integer λ such that a PSCA(v,t,λ) exists. Kuperberg, Lovett and Peled proved g(v,t)=O(v t ) using probabilistic methods. We present an explicit construction that proves g(v,t)=O(v t(t-2) ) for fixed t⩾4. The method of construction involves taking a permutation representation of the group of projectivities of a suitable projective space of dimension t-2 and deleting all but a certain number of symbols from each permutation. In the case that this space is a Desarguesian projective plane, we also show that there exists a permutation representation of the group of projectivities of the plane that covers the vast majority of 4-sequences of its points the same number of times.