Anti-Hadamard Matrices, Coin Weighing, Threshold Gates, and Indecomposable Hypergraphs

Noga Alon, Van H. Vũ

Research output: Contribution to journalArticle

28 Scopus citations

Abstract

Letχ1(n) denote the maximum possible absolute value of an entry of the inverse of annbyninvertible matrix with 0,1 entries. It is proved thatχ1(n)=n(1/2+o(1))n. This solves a problem of Graham and Sloane. Letm(n) denote the maximum possible numbermsuch that given a set ofmcoins out of a collection of coins of two unknown distinct weights, one can decide if all the coins have the same weight or not usingnweighings in a regular balance beam. It is shown thatm(n)=n(1/2+o(1))n. This settles a problem of Kozlov and Vũ. LetD(n) denote the maximum possible degree of a regular multi-hypergraph onnvertices that contains no proper regular nonempty subhypergraph. It is shown thatD(n)=n(1/2+o(1))n. This improves estimates of Shapley, van Lint and Pollak. All these results and several related ones are proved by a similar technique whose main ingredient is an extension of a construction of Håstad of threshold gates that require large weights.

Original languageEnglish (US)
Pages (from-to)133-160
Number of pages28
JournalJournal of Combinatorial Theory. Series A
Volume79
Issue number1
DOIs
StatePublished - Jul 1 1997
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics

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