New upper bounds for nonbinary codes based on the Terwilliger algebra and semidefinite programming

Dion Gijswijt, Alexander Schrijver, Hajime Tanaka

Research output: Contribution to journalArticlepeer-review

37 Citations (Scopus)

Abstract

We give a new upper bound on the maximum size Aq (n, d) of a code of word length n and minimum Hamming distance at least d over the alphabet of q ≥ 3 letters. By block-diagonalizing the Terwilliger algebra of the nonbinary Hamming scheme, the bound can be calculated in time polynomial in n using semidefinite programming. For q = 3, 4, 5 this gives several improved upper bounds for concrete values of n and d. This work builds upon previous results of Schrijver [A. Schrijver, New code upper bounds from the Terwilliger algebra and semidefinite programming, IEEE Trans. Inform. Theory 51 (2005) 2859-2866] on the Terwilliger algebra of the binary Hamming scheme.

Original languageEnglish
Pages (from-to)1719-1731
Number of pages13
JournalJournal of Combinatorial Theory. Series A
Volume113
Issue number8
DOIs
Publication statusPublished - 2006 Nov

Keywords

  • Block-diagonalization
  • Codes
  • Delsarte bound
  • Nonbinary codes
  • Semidefinite programming
  • Terwilliger algebra
  • Upper bounds

ASJC Scopus subject areas

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

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