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New upper bounds for nonbinary codes based on the Terwilliger algebra and semidefinite programming |
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| Authors: | Dion Gijswijt Hajime Tanaka |
| Institution: | a Department of Mathematics, University of Amsterdam, Plantage Muidergracht 24, 1018 TV Amsterdam, The Netherlands b CWI, Kruislaan 413, 1098 SJ Amsterdam, The Netherlands c Division of Mathematics, Graduate School of Information Sciences, Tohoku University, Sendai, Japan |
| 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. |
| Keywords: | Codes Nonbinary codes Upper bounds Delsarte bound Terwilliger algebra Block-diagonalization Semidefinite programming |
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