On Perfect Sequence Covering Arrays

Abstract

A PSCA$$(v, t, \lambda )$$ ( v , t , λ ) is a multiset of permutations of the v-element alphabet $$\{0, \dots , v-1\}$$ { 0 , ⋯ , v - 1 } , such that every sequence of t distinct elements of the alphabet appears in the specified order in exactly $$\lambda $$ λ of the permutations. For $$v \geqslant t \geqslant 2$$ v ⩾ t ⩾ 2 , we define g(v, t) to be the smallest positive integer $$\lambda $$ λ , such that a PSCA$$(v, t, \lambda )$$ ( v , t , λ ) exists. We show that $$g(6, 3) = g(7, 3) = g(7, 4) = 2$$ g ( 6 , 3 ) = g ( 7 , 3 ) = g ( 7 , 4 ) = 2 and $$g(8, 3) = 3$$ 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 $$v \leqslant 32$$ v ⩽ 32 and $$3\leqslant t\leqslant 6$$ 3 ⩽ t ⩽ 6 . We also prove a number of restrictions on the distribution of symbols among the columns of a PSCA.

Growth of the perfect sequence covering array number

“…The optimized recursion leading to (10) permits us to improve on some recently obtained results in [5] (by taking the exact values of the recursion). We state the hereby found bounds in Table 1.…”

“…It remains open by how much lower and upper bounds for g(n, 3) can still be improved. The great difficulty, already for k ∈ {3, 4} and small n, to determine existence of PSCAs (on a certain number of rows) manifests itself in some works, which come up with computationally intensive search procedures [5,11]. It would be highly interesting to find out whether the regularity, that PSCAs have compared to SCAs, facilitates the determination of the complexity class of the problem of calculating g(n, k)k!, the minimum number of rows able to host a PSCA.…”

“…It appears that within the class of matrices having uniform columnwise distribution of symbols (cf. [5]), this latter constraint still allows to find representatives of PSCAs. Indeed, by a recursive search for strength k = 3, appending column to column, we find that such special representatives exist for the parameter constellation (n, k, λ) ∈ {(3, 3, 1), (6, 3, 2)}.…”

“…The representative of PSCA (6,4,1) presented in [10,Proposition 2.22] is reflection symmetric, too. For strength k = 4, the constellation (n, k, λ) = (8, 4, 3) is next, for which it is still open whether it possesses such a special representative (since it was shown in [5] that PSCA(8, 4, 2) = ∅).…”

A polynomial construction of perfect sequence covering arrays

“…In addition to the asymptotic results cited above, research in this area has uncovered exact values of g(v, t) [4,11,12,13,17] as well as the non-existence of a PSCA(v, t, λ) [2,4,8,12] for certain choices of v, t and λ.…”