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 Gröbner basis of such an ideal relative to a degreecompatible 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.2010 Mathematics Subject Classification. Primary: 94B05, 13P25; Secondary: 13P10.

Heuristic decoding of linear codes using commutative algebra

“…The generalized code ideal will be considered next [14]. For this, let α be a primitive element of the field F q .…”

“…The generalized code ideal of a linear code can be used for complete decoding [5,12,14]. The key ingredients are the reduced Gröbner basis with respect to any degree compatible ordering and the division algorithm for multivariate polynomials.…”

“…A generating set for the code ideal I + (C) will contain both a generating set of the associated linear code as well as their scalar multiples and an encoding of the additive structure of the field F q [14,17]. The latter can be given by the ideal …”

“…Moreover, Gröbner bases have been used to compute the minimum distance of a binary linear code. This approach has been extended to linear codes over finite prime fields and integral residue class rings [10,13] and to linear codes over arbitrary finite fields [4,14].…”

Graver Bases and Niversal Grobner Bases for Linear Codes

“…A generating set for the code ideal I + (C) will contain both a generating set of the associated linear code as well as their scalar multiples and an encoding of the additive structure of the field F q [14,17]. The latter can be given by the ideal I q in K[x] generated by the set…”