Growth of the perfect sequence covering array number

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)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g(n,k)$$\end{document} (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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_n$$\end{document} as rows, such that each ordered k-sequence of distinct elements of [n]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[n]$$\end{document} 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.


Introduction
Set [n] = {1, . . ., n}.Let S n denote the symmetric group on [n], and define S n,k as the set of all ordered tuples (x 1 , . . ., x k ) ∈ [n] k whose entries x j are distinct.Following [14], a PSCA of strength k and multiplicity λ ≥ 1 on n symbols is a matrix A ∈ [n] (λk!)×n with distinct symbols per row such that each sequence s ∈ S n,k is contained as a subsequence in exactly λ rows (s is said to be covered λ times by A).Denote the class of all such matrices A by PSCA(n, k, λ).Let g(n, k) be the minimum multiplicity λ such that PSCA(n, k, λ) is non-empty, and call g(n, k) the perfect sequence covering array number (in analogy to [2]).A PSCA can be seen as a (k, λ)-directed design of blocksize n and order n (cf.[4]).PSCAs are particular sequence covering arrays (SCAs), i.e., matrices with permutations as rows that cover each s ∈ S n,k at least once [2].SCAs were introduced in [8] for the purpose of event sequence testing (cf.also [2,4]).
Recently, the following two results were obtained by Yuster.
Theorem 1 [14] If k/2 is a prime, then for all n ≥ k we have For arbitrary k, provided n k, the bound g(n, k) > n k/2−o k (1) applies.
We now point to an isomorphic structure being analyzed in the setting of min-wise independent permutations [3].We index families F ⊂ S n by an index set [d], d ∈ N. The elements are allowed to occur repeatedly, and hence the cardinality of F refers to the cardinality of the indexing set.We need the concept of the rank of an element before we get to the actual definition of interest.
Definition 1 For a subset X ⊂ [n] and arbitrary x ∈ X , we denote the rank of x in X as rank(x, X ) := |{y ∈ X : y < x}|.
when π is drawn uniformly at random from F .
It is clear that (2) corresponds to the following condition: For each sequence (x 1 , . . ., x k ) ∈ S n,k and for each permutation π randomly drawn (uniform probability) from F , we have Next, we give a proposition that illuminates the isomorphy of PSCAs and rankwise independent families.Before that, we need an appropriate mapping and an auxiliary observation from which the proposition follows.For the sake of completeness we include a proof of the following Lemma 3 which was already noticed in [4,Lemma 1.1].
Conversely, if there is λ ≥ 1 such that A(F ) ∈ PSCA(n, k, λ), then for x ∈ S n,k there are λ rows in A(F ) that cover x.By Lemma 3, F must contain λ permutations which all are in C x (here permutations are counted with respect to multiplicity).Therefore, a permutation π chosen uniformly at random from The inverse operation of A(•) in Proposition 4, i.e., the conversion of a PSCA to a rankwise independent family, is determined by interpreting the rows of the PSCA as permutations and successively storing their inverses one by one in a family of permutations.
In [6] it is already remarked that k-rankwise independence implies -rankwise independence, for all ∈ [k].In other words, a nesting property analogous to the one for PSCAs holds (if A ∈ PSCA(n, k, λ), then also A ∈ PSCA(n, , λk!/ !) [14]).We highlight that k-rankwise independent families can be considered as completely scrambling families1 (introduced by Spencer [12]) with an additional requirement of regularity.Furthermore, k-rankwise independence is a special case of k-restricted min-wise independence (cf.Definition 3).It is known (cf.[6, p. 139]) that these two notions coincide for k = 3 and that k-rankwise independence is strictly more specific when k > 3.In [3], k-restricted min-wise independent families were introduced to efficiently estimate the resemblance of two documents.The latter is motivated by practice, where the aim is to reduce the computational cost of searching for near-duplicate documents on the World Wide Web [3].
Remark 1 Given a matrix in PSCA(n, k, λ) with n ≥ k + 1, we can utilize it to construct a matrix in PSCA(n − 1, k, λ) by dropping the symbol n in each row (see [14]).Consequently, we also know how to construct a k-rankwise independent family F ⊂ S n−1 (with |F | = |F |) from a k-rankwise independent family F ⊂ S n (this was already noticed and applied in [6] without referring to PSCAs).
We can now transfer bounds from rankwise independent families to PSCAs (and vice versa).
In this section asymptotic results for rankwise independent permutations will be stated jointly with their implications (cf.Proposition 4) for the number g(n, k).
The following result can in particular be analyzed when h is a prime; we can therefore compare it with Theorem 1 and conclude that it improves the lower bound by the factor !h, the subfactorial of h,2 being superexponential in h, but independent of n.
Theorem 5 [1] Let F ⊂ S n be a k-rankwise independent family.If k = 2h (h ∈ N), then we have The next Theorem 6 follows from the proof of [6, Theorem 3.2].It positively answers the question of Yuster on the existence of non-trivial upper bounds for g(n, k) for fixed k.In fact, the estimate is polynomial in n.
i=1 p e i i under the side constraints that (k − 1)! divides Q, and p e i i > p, for i = 1, . . ., m.Then, there exists a k-rankwise independent family P of permutations of [ p] such that |P| ≤ ( p k − p)Q k/2 .Consequently, cf.Remark 1, there exists a family P ⊂ S n satisfying Remark 2 For the case that (k − 1)! > n (not covered by Theorem 6), it is established in [6] that a k-rankwise independent family with cardinality of order e O(k 3 ) exists.

Remark 3
The proof of Theorem 6 presented in [6] is constructive: For a prime p and the finite field F p of p elements (F s is to be understood accordingly, when s is a prime power), the set of univariate polynomials is used as base for the construction.Under the assumptions of Theorem 6, setting a tuple (b y ) y∈F p ∈ {0, . . ., Q − 1} p is generated by a manipulation which depends on a fixed parameter h = (h 1 , . . ., h m ) ∈ E. The tuple (b y ) y is then used to carry out the so-called " k/2 -tie breaking scheme" [6] which serves to identify each of the p k − p elements of F p (ξ, [1, k]) with a permutation of {0, . . ., p − 1}.Deriving from each tuple in E (having ) such a collection of permutations, yields a total of ( p k − p)Q k/2 permutations.Collectively these permutations fulfill krankwise independence.For a more detailed discussion of the proof of Theorem 6 (and of the aforementioned scheme) we point to [6].We state more specific results for k ∈ {3, 4}.Originally, the subsequent result concerning 3-rankwise independence was established for the equivalent property of 3-restricted min-wise independence.
Theorem 7 ([13]) Let n ≥ 4.Then, there exists a 3-rankwise independent family E ⊂ S n and a 4-rankwise independent family F ⊂ S n , with cardinalities g(n, 3) Remark 4 The latter result is obtained by methods from affine/projective finite geometry by a recursive construction.The bound (8) improves on the bound in Theorem 2 by a factor of approximately log 2 (n) 0.81 .The bound ( 9) is obtained by estimating the members of a recursive sequence which describes cardinalities of 4-rankwise independent families of permutations of [n], for n = 2 2q +1.The involved recursion is homogeneous in the cardinality of the family employed as base case.For the construction it is required to start with a 4-rankwise independent family of F ⊂ S 5 .In [13], the authors choose F = S 5 , such that |F | = 120.We can improve this base case by making use of a construction due to Levenshtein [9] allowing to construct a PSCA(n +1, n, 1).It provides us with a PSCA(5, 4, 1), which we can map via the isomorphism in Proposition 4 to a 4-rankwise independent family F satisfying | F | = 24.Hence, we obtain an improvement of the bound (9), which reads 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. a 3-rankwise independent family of cardinality 6 has been used (such a family exists due to PSCA(4, 3, 1) = ∅, cf.[9]).The bound slightly improving upon (8), is obtained.By evaluation of the respective recursion, bounds for g(15, 3) and g(16, 3) have been improved (see Table 2).

Conclusion and open questions
In our discussion, we have found that the theory of PSCAs and the theory of min-wise independent permutations can benefit considerably from each other.Despite the isomorphy of many concepts in these theories, strong interconnections seemed not to be exploited in the past (perhaps because one theory consistently uses probabilistic language).By combining the latter theories, we have improved several bounds and established polynomial boundedness for g(n, k), which Yuster asked for in [14].Furthermore, we achieved progress in another question appearing in [14] which asks to find the right order of magnitude of g(n, 3): The quasi-linear upper bound (established in [14]) is tightened by a factor of log 2 (n) 0.81 (cf.( 8

)).
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.For SCAs the respective problem concerning the minimal row count is still open (even if in [2] NP-completeness has been established for an altered problem).
Aiming to calculate further exact values of g(n, k), we experimentally tried, for n being a divisor of λk!, to seek small PSCAs by restricting the search space to a class of matrices A (having permutations as rows) and satisfying the following "reflection symmetry" property: For any column indices j 1 < j 2 , the ( j 1 , j 2 )-submatrix of A satisfies that if h rows coincide with (a, b) ∈ [n] 2 , then also h rows coincide with (b, a).The motivation behind this additional assumption is that recently PSCAs being unions of cosets of S n have successfully been found (cf.[11]) for several parameter constellations (n, k, λ).We suspect that our proposed kind of symmetry induces a suitable balancing of symbols being advantageous for finding PSCAs, too.
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)}.Existence for the constellation (9,3,3) could no be determined due to computational limitations and we would find it interesting to investigate this case with more dedicated computational effort.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) = ∅).
Combinatorial aspects (e.g.enumeration for small n and efficient constructions) of the class of matrices satisfying the latter reflection symmetry might be of independent interest.