Distribution of the Minimum Distance of Random Linear Codes

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 (
superpolynomially
in
$n$
) 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
$2< q< 49$
.