I'm quite interested in the following algorithmic problem, on which I can't find any information. Phrased as a decision problem:
Given a set of vectors $V$ in $\text{GF}(2)^n$, a vector $\mathbf u$ in $\text{GF}(2)^n$, and an integer $k$, decide whether there exists a subset $S \subseteq V$ such that $|S|\leq k$ and $\sum_{\mathbf v\in S} \mathbf v = \mathbf u$.
Note that despite sounding similar to the subset sum problem, it's actually quite different – in particular without the $|S|\leq k$ restriction it's basic linear algebra. My attempts to google the problem have failed so far. Is this a problem already explored in some papers? Is it equivalent to another well-known problem? Is it solvable by a practical algorithm? Is it NP-complete? I would be surprised if there was no literature about this problem at all given the short formulation.
What I figured out by myself is that this problem could be polynomially reduced to the problem of
Given a set of vectors $V$ in $\text{GF}(2)^n$, a vector $\mathbf u$ in $\text{GF}(2)^n$, and an integer $k$, decide whether there exists a subset $S \subseteq V$ such that the vector $\mathbf u+\sum_{\mathbf v\in S} \mathbf v$ consists of at most $k$ ones.
Maybe this one is better-known?
