Existence of matrices with some invertibility properties Prove that there exists five  matrices $B_i \in \mathbb{F}_2^{5\times 10}$, $i\in \{1,2,3,4,5\}$, such that any two $B_i$'s form an invertible matrix in $\mathbb{F}_2^{10\times 10}$.
I am interested in a proof of existence that could be generalized to matrices with other dimensions.
Follow-up question when there are constraints on the $B_i$'s:  Existence of matrices in $\mathbb{F}_2$ with some invertibility properties
 A: If $B$, $B'$ are $5\times 10$ matrices of rank $5$ over $\mathbb{F}_2$, the condition that the matrix formed by stacking $B$ on top of $B'$ is invertible is equivalent to the condition that the row span of $B$ and the row span of $B'$ have trivial intersection. The row spans are $5$-dimensional subspaces of $\mathbb{F}_2^{10}$, so given five $5$-dimensional subspaces of $\mathbb{F}_2^{10}$ with pairwise trivial intersection, we can choose bases for them to construct matrices $B_1,\ldots,B_5$.
Note that $\mathbb{F}_2^{10}$ is isomorphic to $\mathbb{F}_{2^5}^2$ as an $\mathbb{F}_2$ vector space. A $1$-dimensional $\mathbb{F}_{2^5}$-subspace of $\mathbb{F}_{2^5}^2$ can be identified with a $5$-dimensional $\mathbb{F}_2$ subspace of $\mathbb{F}_2^{10}$, and every pair of $1$-dimensional $\mathbb{F}_{2^5}$-subpsaces of $\mathbb{F}_{2^5}^2$ have trivial intersection. There are 33 such subspaces, so we can find even find $B_1,\ldots,B_{33}\in\mathbb{F}_2^{5\times 10}$ with the desired property.
