  We consider the problem of determining Gr\"obner bases of binomial ideals
associated with linear error correcting codes. Computation of Gr\"obner 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, Gr\"obner bases of linear codes associated to Grassmann
varieties and Schubert varieties over a binary field have been obtained. We
also use them to study the decoding of binary Schubert codes.
