Let $S_1, \ldots, S_m \in \mathbb{Z}_2^n$ be $n$-tuples forming a subgroup of $\mathbb{Z}_2^n$. How can I find a set of congruences (mod 2) over $x_1, \ldots, x_n$ satisfying exactly $x_1 = S_j[1], \ldots, x_n = S_j[n]$ for all $j \in [m]$.

For exemple, given $S_1 = 000$ and $S_2 = 111$, this is an associated set of congruences: $$x_1 \equiv x_2 \, (\text{mod 2}) \text{ and } x_3 \equiv x_2 \, (\text{mod 2}).$$

I've also post the question on math.stackexchange, sorry for the inconvenience.