Graver Bases and Universal Gröbner Bases for Linear Codes

نویسندگان

  • Natalia Dück
  • Karl-Heinz Zimmermann
چکیده

Linear codes over any finite field can be associated to binomial ideals. Focussing on two specific instances, called the code ideal and the generalized code ideal, we present how these binomial ideals can be computed from toric ideals by substituting some variables. Drawing on this result we further show how their Graver bases as well as their universal Gröbner bases can be computed.

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

On the Universal Gröbner Bases of Varieties of Minimal Degree

A universal Gröbner basis of an ideal is the union of all its reduced Gröbner bases. It is contained in the Graver basis, the set of all primitive elements. Obtaining an explicit description of either of these sets, or even a sharp degree bound for their elements, is a nontrivial task. In their ’95 paper, Graham, Diaconis and Sturmfels give a nice combinatorial description of the Graver basis f...

متن کامل

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 integ...

متن کامل

On the universal Gröbner bases of toric ideals of graphs

To any graph G can be associated a toric ideal I G. In this talk some recent joint work with Enrique Reyes and Christos Tatakis will be presented on the toric ideal of a graph. A characterization in graph theoretical terms of the elements of the Graver basis and the universal Gröbner basis of the toric ideal of a graph will be given. The Graver basis is the union of the primitive binomials of t...

متن کامل

Gröbner Bases for Schubert Codes

We consider the problem of determining Gröbner bases of binomial ideals associated with linear error correcting codes. Computation of Gröbner bases of linear codes have become a topic of interest to many researchers in coding theory because of its several applications in decoding and error corrections. In this paper, Gröbner bases of linear codes associated to Grassmann varieties and Schubert v...

متن کامل

Exploiting Symmetries in the Computation of Graver Bases

Many challenging Graver bases computations, like for multi-way tables in statistics, have a highly symmetric problem structure that is not exploited so far computationally. In this paper we present a Graver basis algorithm for sublattices of Zn that exploits existing symmetry.

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

عنوان ژورنال:
  • CoRR

دوره abs/1401.6304  شماره 

صفحات  -

تاریخ انتشار 2014