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?



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


