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 number$$g(n,k)$$ g ( 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 $$S_n$$ S n as rows, such that each ordered k-sequence of distinct elements of $$[n]$$ [ 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.

A polynomial construction of perfect sequence covering arrays

“…Although this result gives a tighter bound on g(v, t) than Theorem 1.1, it is not yet known how to efficiently construct either a PSCA or a t-wise uniform set of permutations with this size. Recently, Iurlano [7] has established an equivalence between PSCAs of strength t and families of t-rankwise independent permutations. Iurlano also uses a construction of Itoh, Takei and Tarui [6] of t-rankwise independent permutations to build PSCAs with v O(t 2 /ln t) permutations.…”