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- W2965196348 abstract "We say a subset C subseteq {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 (log_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 Komlos 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 present a method to obtain the first improvement to the Fredman-Komlos bound for every k >= 5, and we apply this method to give explicit numerical bounds for k=5, 6." @default.
- W2965196348 created "2019-08-13" @default.
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- W2965196348 date "2018-01-01" @default.
- W2965196348 modified "2023-09-23" @default.
- W2965196348 title "Beating Fredman-Komlós for perfect k-hashing." @default.
- W2965196348 hasPublicationYear "2018" @default.
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