GR{\"O}BNER BASES FOR PERFECT BINARY LINEAR CODES
Natalia Dück, K.-H. Zimmermann · International Journal of Pure and Apllied Mathematics · 2014
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.