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Beating Fredman-Komlós for Perfect k-Hashing

Authors Venkatesan Guruswami , Andrii Riazanov

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Venkatesan Guruswami
  • Computer Science Department, Carnegie Mellon University, 5000 Forbes Ave, Pittsburgh, PA, USA, 15213
Andrii Riazanov
  • Computer Science Department, Carnegie Mellon University, 5000 Forbes Ave, Pittsburgh, PA, USA, 15213


Some of this work was done when the first author was visiting the School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore and the Center of Mathematical Sciences and Applications, Harvard University.

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Venkatesan Guruswami and Andrii Riazanov. Beating Fredman-Komlós for Perfect k-Hashing. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 92:1-92:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Error-correcting codes
  • Mathematics of computing → Coding theory
  • Coding theory
  • perfect hashing
  • hash family
  • graph entropy
  • zero-error information theory


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