Parity check matrices and product representations of squares

Assaf Naor, Jacques Verstraëte

Research output: Contribution to journalArticle

Abstract

Let NF (n,k,r) denote the maximum number of columns in an n-row matrix with entries in a finite field NF in which each column has at most r nonzero entries and every k columns are linearly independent over N F. We obtain near-optimal upper bounds for NF (n,k,r) in the case k > r. Namely, we show that NF (n,k,r) ≫ n r/2 + cr/k where c ≈ 4/3 for large k. Our method is based on a novel reduction of the problem to the extremal problem for cycles in graphs, and yields a fast algorithm for finding short linear dependencies. We present additional applications of this method to a problem on hypergraphs and a problem in combinatorial number theory.

Original languageEnglish (US)
Pages (from-to)163-185
Number of pages23
JournalCombinatorica
Volume28
Issue number2
DOIs
StatePublished - Mar 2008

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Computational Mathematics

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