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

Authors:
Dion Gijswijt;Alexander Schrijver;Hajime Tanaka
Affiliations:
Department of Mathematics, University of Amsterdam, TV Amsterdam, The Netherlands;CWI, Kruislaan, SJ Amsterdam, The Netherlands and Department of Mathematics, University of Amsterdam, TV Amsterdam, The Netherlands;Division of Mathematics, Graduate School of Information Sciences, Tohoku University, Sendai, Japan
Venue:
Journal of Combinatorial Theory Series A - Special issue in honor of Jacobus H. van Lint
Year:
2006

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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 Schrijvcr [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.