IJPAM: Volume 86, No. 2 (2013)

UNIVERSAL GRÖBNER BASES FOR BINARY LINEAR CODES

Natalia Dück1, Karl-Heinz Zimmermann2
1,2Hamburg University of Technology
21073, Hamburg, GERMANY


Abstract. 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.

Received: February 7, 2013

AMS Subject Classification: 13P10, 94B05

Key Words and Phrases: linear code, Gröbner basis, universal Gröbner basis, Graver basis, circuit, toric ideal, Singleton code

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DOI: 10.12732/ijpam.v86i2.9 How to cite this paper?
Source:
International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2013
Volume: 86
Issue: 2