Further strengthening of upper bounds for perfect k-Hashing
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For a fixed integer k, a problem of relevant interest in computer science and combinatorics is that of determining the asymptotic growth, with n, of the largest set for which a perfect k-hash family of n functions exists. Equivalently, determining the asymptotic growth of a largest subset of {1,2,…,k}^n such that for any k distinct elements in the set, there is a coordinate where they all differ. An important asymptotic upper bound for general k was derived by Fredman and Komlós in the '80s. Only very recently this was improved for general k by Guruswami and Riazanov while stronger results for small values of k were obtained by Arikan, by Dalai, Guruswami and Radhakrishnan and by Dalai and Costa. In this paper, we further improve the bounds for 5≤ k ≤ 8. The method we use, which depends on the reduction of an optimization problem to a finite number of cases, shows that further results might be obtained by refined arguments at the expense of higher complexity.
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