Growth of the perfect sequence covering array number

Abstract

In this note we answer positively an open question posed by Yuster in 2020 [14] on the polynomial boundedness of the perfect sequence covering array numberg(n, k) (PSCA number). The latter determines the (renormalized) minimum row-count that perfect sequence covering arrays (PSCAs) can possess. PSCAs are matrices with permutations in Sn as rows, such that each ordered k-sequence of distinct elements of [n] is covered by the same number of rows. We obtain the result after illuminating an isomorphism between this structure from design theory and a special case of min-wise independent permutations. Afterwards, we point out that asymptotic bounds and constructions can be transferred between these two structures. Moreover, we sharpen asymptotic lower bounds for g(n, k) and improve upper bounds for g(n, 4) and g(n, 3), for some concrete values of n. We conclude with some open questions and propose a new matrix class being potentially advantageous for searching PSCAs.