Abstract. There is a deep connection between linear codes and combinatorial designs. Combinatorial designs can give rise to linear codes and vice versa. In particular, perfect codes always hold combinatorial designs. Recently, linear codes have been associated to binomial ideals by the so-called code ideal which completely describes the code. It will be shown that for a perfect binary linear code, the codewords of minimum Hamming weight are in one-to-one correspondence with the elements of a reduced Gröbner basis for the code ideal with respect to any graded order.

Received: April 17, 2013

AMS Subject Classification: 13P10, 94B05, 51E10

Key Words and Phrases: Gröbner basis, linear code, perfect code, Steiner system, minimum distance

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DOI: 10.12732/ijpam.v91i2.2 How to cite this paper?
Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2014
Volume: 91
Issue: 2

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