When working in $\mathbb R^n$, there is a vast amount of literature on the study of solving $Ax=b$ as well as minimizing $c^T x$ subject to $Ax =b$. I'm thinking of Dantzig-Wolfe decomposition, Farkas's lemma, Caratheodory's theorem and all that kind of stuff. I'm wondering where to search for similar results with respect to solving/decomposing/optimizing over $Ax=b$ in finite fields.
Specifically, I am curious about methods on parallelizing solutions to systems of linear equations via decomposition techniques, rather than message-passing.