# Existence of matrices in the field $\mathbb{F}_2$ with some invertibility properties

All the matrices in this statement are in the field $$\mathbb{F}_2$$. Let $$I$$ be the identity matrix of size $$10 \times 10$$ and let $$e_1$$, $$e_2$$, $$\ldots$$, $$e_{10}$$ denote its rows. For $$i\in \{1,5 \}$$, define the $$2 \times 10$$ matrix $$A_{i} = \left( \begin{matrix} e_{2i-1} \\ e_{2i}\end{matrix} \right)$$.

Without a computer-aided method, can one prove that there exist five matrices $$X_i$$, $$i\in\{1,5\}$$, with size $$3 \times 10$$ such that $$\forall i\in \{1,5\},\forall j\in \{1,5\}\backslash\{i\}, \left( \begin{matrix} X_i \\ X_j \\ A_i \\A_j \end{matrix}\right)$$ is a $$10\times 10$$ invertible matrix?

You may set the rows of $$X_i$$ to be $$e_{2i-1-k}+e_{2i+k}$$, for $$k=1,2,3$$. Due to symmetry, there are only two cases to check, and both work.
• @Ilya Bogdanov How did you come up with this solution? For instance, I see that it makes sense that the $2(i-1)+1$-th and $2i$-th columns of $X_i$ can be set to zero (because $X_i$ is always considered with $A_i$), and that among the five $j$-th columns of the $X_i$'s only three need to be non-zero (because we consider two $X_i$'s and the corresponding $A_i$'s). Which other constraints or other reasonings did you use?