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In this paper, we study the distribution of the minimal distance (in theHamming metric) of a random linear code of dimension $k$ in $\mathbb{F}_q^n$.We provide quantitative estimates showing that the distribution function of theminimal distance is close ({\it{}superpolynomially} in $n$)to the cumulativedistribution function of the minimum of $(q^k-1)/(q-1)$ independent binomialrandom 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 sensethat apart from identification of the weights of proportional codewords, theprobabilistic dependencies introduced by the linear structure of the randomcode, produce a negligible effect on the minimal code weight. As a corollary ofthe main result, we obtain an improvement of the Gilbert--Varshamov bound for$2<q<49$.

Distribution of the minimal distance of random linear codes