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You can get a square-root speed-up with a time-space tradeoff if you are working in $\mathbb{F}_2$. The matrix $M$ is a no-instance iff there exists a non-zero vector $v$ of Hamming weight $\le k$ (i.e., with at most $k$ non-zero coordinates) such that $Mv=0$. This is equivalent to saying that there exist $t,u$ of Hamming weight $\le k/2$ such that $Mt=Mu$ (...

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