  Recently, it was shown that a binary linear code can be associated to a
binomial ideal given as the sum of a toric ideal and a non-prime ideal. Since
then two different generalizations have been provided which coincide for the
binary case. In this paper, we establish some connections between the two
approaches. In particular, we show that the corresponding code ideals are
related by elimination. Finally, a new heuristic decoding method for linear
codes over prime fields is discussed using Gr\"obner bases.
