This question relates to this post, which is ultimately concerned with this paper. I do my best to relay the relevant information here.
Within the linked paper, the authors seem to work under the assumption that if we select a random $288 \times 216$ matrix $M$ with entries in $GF(2)$ (the field with two elements), then there is a very high probability that $M$ has full column rank, and that a randomly selected $84 \times 216$ submatrix will have full column rank. With that in mind, my question is the following:
If a random $m \times n$ matrix $M$ over $GF(2)$ is selected (i.i.d. uniformly random $\{0,1\}$ entries), then what is the probability that $M$ will have maximal rank in the case where
- $m = 288, n =216$
- $m = 84, n = 216$
I am aware that there is a nice formula for this probability in the case where $M$ is square, but I'm not sure where to find (or how to derive off the top of my head) the corresponding probability for rectangular matrices.
Any help is appreciated.
Following the formula given here, we find that the answer to 1 is $$ \prod_{i=1}^{216}(1 - 2^{i-1-288}) \approx 1 - 2\times 10^{-22}, $$ and the answer to 2 is $$ \prod_{i=1}^{84}(1 - 2^{i-1-216}) \approx 1 - 2\times 10^{-40}. $$