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New upper bounds for nonbinary codes based on the Terwilliger algebra and semidefinite programming. (English) Zbl 1105.94027
Summary: We give a new upper bound on the maximum size \(A_{q}(n,d)\) of a code of word length \(n\) and minimum Hamming distance at least \(d\) over the alphabet of \(q\geq 3 \) 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 A. Schrijver [IEEE Trans. Inf. Theory 51, 2859–2866 (2005)] on the Terwilliger algebra of the binary Hamming scheme.

MSC:
94B65 Bounds on codes
05E30 Association schemes, strongly regular graphs
90C22 Semidefinite programming
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References:
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