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Range-compatible homomorphisms over fields with two elements

Let $U$ and $V$ be finite-dimensional vector spaces over a (commutative) field $\mathbb{K}$, and $\mathcal{S}$ be a linear subspace of the space $\mathcal{L}(U,V)$ of all linear operators from $U$ to $V$. A map $F : \mathcal{S} \rightarrow V$ is called range-compatible when $F(s) \in \text{Im}(s)$ for all $s \in \mathcal{S}$. In a previous work, we have classified all the range-compatible group homomorphisms provided that $\text{codim}(\mathcal{S}) \leq 2\,\text{dim}(V)-3$, except in the special case when $\mathbb{K}$ has only two elements and $\text{codim}(\mathcal{S}) = 2\,\text{dim}(V)-3$. In this article, we give a thorough treatment of that special case. Our results are partly based upon the recent classification of vector spaces of matrices with rank at most $2$ over $\mathbb{F}_2$. As an application, we classify the $2$-dimensional non-reflexive operator spaces over any field, and the affine subspaces of $\text{M}_{n,p}(\mathbb{K})$ with lower-rank $2$ and codimension $3$.