Distribution of the minimal distance of random linear codes

TL;DR

This paper analyzes the distribution of the minimal Hamming distance in random linear codes, showing it closely resembles the minimum of independent binomial variables and converges to a Gumbel distribution, with implications for bounds on code parameters.

Contribution

It provides a precise probabilistic characterization of the minimal distance distribution in random linear codes, confirming the negligible effect of linear dependencies.

Findings

Distribution closely matches the minimum of independent binomial variables.

Distribution converges to a Gumbel distribution at integer points.

Improves the Gilbert-Varshamov bound for certain q values.

Abstract

In this paper, we study the distribution of the minimal 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 minimal distance is close ({\it{}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 minimal code weight. As a corollary of the main result, we obtain an improvement of the Gilbert--Varshamov bound for $2<q<49$.