  In this paper we introduce a binomial ideal derived from a binary linear
code. We present some applications of a Gr\"obner basis of this ideal with
respect to a total degree ordering. In the first application we give a decoding
method for the code. By associating the code with the set of cycles in a graph,
we can solve the problem of finding all codewords of minimal length (minimal
cycles in a graph), and show how to find a minimal cycle basis. Finally we
discuss some results on the computation of the Gr\"obner basis.
