Applications of estimates of the probability that a random $k$-dimensional subspace is of minimal weight

We find nontrivial estimates of the probability that a random $k$-dimensional subspace of an $n$-dimensional vector space over a finite field $GF(q)$ is of minimal weight. The conditions are $nq^{k-n} \leq 1$ in Theorem 1 and $k \geq n-k \geq 4$ in Theorem 2. Some applications of the estimates for finding the asymptotic behavior of the above probability are given.