Decomposition of Modular Codes for Computing Test Sets and Graver Basis

Abstract

In order to obtain the set of codewords of minimal support for codes defined over \({\mathbb{Z}_q}\), one can compute a Graver basis of the ideal associated to such codes. The main aim of this article is to reduce the complexity of the algorithm obtained by the authors in a previous work taking advantage of the powerful decomposition theory for linear codes provided by the decomposition theory of representable matroids over finite fields. In this way we identify the codes that can be written as “gluing” of codes of shorter length. If this decomposition verifies certain properties then computing the set of codewords of minimal support in each code appearing in the decomposition is equivalent to computing the set of codewords of minimal support for the original code. Moreover, these computations are independent of each other, thus they can be carried out in parallel for each component, thereby not only obtaining a reduction of the complexity of the algorithm but also decreasing the time needed to process it.