Let$q\geq 2$be a prime power. In this paper, we study the distribution of the minimum distance (in the Hamming metric) of a random linear code of dimension$k$in$\mathbb {F}_{q}^{n}$. We provide quantitative estimates showing that the distribution function of the minimum distance is close (superpolynomiallyin$n$) to the cumulative distribution function of the minimum of$(q^{k}-1)/(q-1)$independent binomial random variables with parameters$\frac {1}{q}$and$n$. The latter, in turn, converges to a Gumbel distribution at integer points when$\frac {k}{n}$converges to a fixed number in (0, 1). Our result confirms in a strong sense that apart from identification of the weights of proportional codewords, the probabilistic dependencies introduced by the linear structure of the random code, produce a negligible effect on the minimum code weight. As a corollary of the main result, we obtain an improvement of the Gilbert–Varshamov bound for$2< q< 49$.