2012/06/22 by Márquez-Corbella, Irene, Martínez-Moro, Edgar, Suárez-Canedo, Emilio · 1 citation
#13P10 #94B05 #Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT)
paper · doi:10.48550/arxiv.1206.5124
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 Gröbner 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.