2026-07-21T12:35:04Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:10668 2024-03-06T10:46:25Z ddc:004 open_access

Beating Fredman-Komlós for Perfect k-Hashing Guruswami, Venkatesan Riazanov, Andrii Coding theory perfect hashing hash family graph entropy zero-error information theory 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 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 present a method to obtain the first improvement to the Fredman-Komlós bound for every k >= 5, and we apply this method to give explicit numerical bounds for k=5, 6. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Venkatesan Guruswami and Andrii Riazanov 2019 Is Part Of LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ICALP.2019.92 urn:nbn:de:0030-drops-106687 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.92 eng https://creativecommons.org/licenses/by/3.0/legalcode