Suppose we have a linear system $Ax=b$ in GF(2), where $A$ is sparse. Its solution is $x = Tz + x_0$.
It is apparent that $T$ is not unique, then is there a way to find a T that is as sparse as possible?
2026-04-08 12:40:57.1775652057
Can the null space of a sparse matrix have a set of basis that is sparse?
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I'm assuming the number $n$ of columns of $T$ is fixed at the dimension of the null space of $A$ (otherwise we could cheat by adding extra columns of $0$'s to $T$). Thus the columns of $T$ are a basis of the null space of $A$ ($x_0$ and $b$ are irrelevant).
If $T_1$ is one possible $T$, the others are $T_1 S$ where $S$ is an arbitrary invertible $n \times n$ matrix. Since there are only finitely many invertible $n \times n$ matrices over $GF(2)$, one (not particularly efficient) algorithm is to enumerate all possibilities.