Distribution of the minimal distance of random linear codes
(1912.12833)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 $\mathbb{F}_qn$. 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 $(qk-1)/(q-1)$ 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 $2<q<49$.
We're not able to analyze this paper right now due to high demand.
Please check back later (sorry!).
Generate a summary of this paper on our Pro plan:
We ran into a problem analyzing this paper.