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Journal Articles Advances in Mathematics of Communications Year : 2015

On the ideal associated to a linear code

Abstract

This article aims to explore the bridge between the algebraic structure of a linear code and the complete decoding process. To this end, we associate a specific binomial ideal I+(C) to an arbitrary linear code. The binomials involved in the reduced Gr\"obner basis of such an ideal relative to a degree-compatible ordering induce a uniquely defined test-set for the code, and this allows the description of a Hamming metric decoding procedure. Moreover, the binomials involved in the Graver basis of I+(C) provide a universal test-set which turns out to be a set containing the set of codewords of minimal support of the code.

Dates and versions

hal-01243389 , version 1 (15-12-2015)

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Irene Márquez-Corbella, Edgar Martinez-Moro, Suárez-Canedo Emilio. On the ideal associated to a linear code. Advances in Mathematics of Communications, 2015. ⟨hal-01243389⟩

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