Universal Gr\"obner Bases for Binary Linear Codes

Each linear code can be described by a code ideal given as the sum of a toric ideal and a non-prime ideal. In this way, several concepts from the theory of toric ideals can be translated into the setting of code ideals. It will be shown that after adjusting some of these concepts, the same inclusion relationship between the set of circuits, the universal Gr\"obner basis and the Graver basis holds. Furthermore, in the case of binary linear codes, the universal Gr\"obner basis will consist of all binomials which correspond to codewords that satisfy the Singleton bound and a particular rank condition. This will give rise to a new class of binary linear codes denoted as Singleton codes.Comment: Accepted for publication in IJPA

Graver Bases and Universal Gr\"obner Bases for Linear Codes

Two correspondences have been provided that associate any linear code over a finite field with a binomial ideal. In this paper, algorithms for computing their Graver bases and universal Gr\"obner bases are given. To this end, a connection between these binomial ideals and toric ideals will be established.Comment: 18 page

On the fan associated to a linear code

International audienceWe will show how one can compute all reduced Gröbner bases with re-spect to a degree compatible ordering for code ideals -even though these binomial ideals are not toric. To this end, the correspondence of linear codes and binomial ideals will be briefly described as well as their resemblance to toric ideals. Finally, we will hint at applications of the degree compatible Gröbner fan to the code equivalence problem

On the fan associated to a linear code

International audienceWe will show how one can compute all reduced Gröbner bases with re-spect to a degree compatible ordering for code ideals -even though these binomial ideals are not toric. To this end, the correspondence of linear codes and binomial ideals will be briefly described as well as their resemblance to toric ideals. Finally, we will hint at applications of the degree compatible Gröbner fan to the code equivalence problem

Graver bases and niversal Gröbner bases for linear codes

A linear code over a finite field can be associated with two binomial ideals. In this paper, algorithms for computing their Graver bases and universal Gröbner bases are given

Model-based Decision Support in Manufacturing and Service Networks

In this paper, we sketch some of the challenges that should be addressed in future research efforts for model- based decision support in manufactur- ing and service networks. This includes integration issues, taking into account the autonomy of the decision-making entities in face of information asymme- try, the modeling of preferences of the decision makers, efficiently determin- ing robust solutions, i.e. solutions that are insensitive with respect to changes in the problem data, and a reduction of the time needed for model build- ing and usage. The problem solution cycle includes problem analysis, the design of appropriate algorithms and their performance assessment. We are interested in a prototypical integration of the proposed methods within appli- cation systems, which can be followed up with field tests of the extended ap- plication systems. We argue that the described research agenda requires the interdisciplinary collaboration of busi- ness and information systems engi- neering researchers with colleagues from management science, computer science, and operations research. In addition, we present some exemplify- ing, illustrative examples of relevant research results