Media type: Electronic Resource; E-Article Title: Universal Gröbner bases for binary linear codes Contributor: Dück, Natalia [Author]; Zimmermann, Karl-Heinz [Author] Published: Academic Publishing, 2013 Language: English ISSN: 1314-3395 Keywords: Graver basis ; linear code ; Gröbner basis ; Singleton code ; circuit ; universal Gröbner basis ; toric ideal Origination: Footnote: Diese Datenquelle enthält auch Bestandsnachweise, die nicht zu einem Volltext führen. Description: 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 Gröbner basis and the Graver basis holds. Furthermore, in the case of binary linear codes, the universal Gröbner 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. © 2013 Academic Publications, Ltd.