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.

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: https://mathoverflow.net/questions/405296/existence-of-matrices-in-mathbbf-2-with-some-invertibility-properties