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
