Beating Fredman-Komlós for perfect k-hashing
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We say a subset C ⊆{1,2,...,k}^n is a k-hash code (also called k-separated) if for every subset of k codewords from C, there exists a coordinate where all these codewords have distinct values. Understanding the largest possible rate (in bits), defined as (_2 |C|)/n, of a k-hash code is a classical problem. It arises in two equivalent contexts: (i) the smallest size possible for a perfect hash family that maps a universe of N elements into {1,2,...,k}, and (ii) the zero-error capacity for decoding with lists of size less than k for a certain combinatorial channel. A general upper bound of k!/k^k-1 on the rate of a k-hash code (in the limit of large n) was obtained by Fredman and Komlós in 1984 for any k ≥ 4. While better bounds have been obtained for k=4, their original bound has remained the best known for each k > 5. In this work, we obtain the first improvement to the Fredman-Komlós bound for every k > 5. While we get explicit (numerical) bounds for k=5,6, for larger k we only show that the FK bound can be improved by a positive, but unspecified, amount. Under a conjecture on the optimum value of a certain polynomial optimization problem over the simplex, our methods allow an effective bound to be computed for every k.
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