Algebraic structure of the minimal support codewords set of some linear codes

It has been widely known that complete decoding for binary linear codes can be regarded as an linear integer programming problem with binary arithmetic conditions. Conti and Traverso [8] have proposed an efficient algorithm which uses Gröbner bases to solve integer programming with ordinary integer arithmetic conditions. Ikegami and Kaji [11] extended the Conti-Traverso algorithm to solve integer programming with modulo arithmetic conditions. It is natural to consider for those problems the Graver basis associated to them which turns to be the minimal cycles of the matroid associated to the code, i.e. minimal support codewords in the binary case and its geometry. This provides us an universal test set for the programs considered.