On the ideal associated to a linear code

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 $I_{+}(\mathcal{C})$ 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 $I_{+}(\mathcal{C})$ provide a universal test-set which turns out to be a set containing the set of codewords of minimal support of the code.