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$2< q< 49$.

Graph matching, also known as network alignment, refers to finding a bijection between the vertex sets of two given graphs so as to maximally align their edges. This fundamental computational problem arises frequently in multiple fields such as computer vision and biology. Recently, there has been a plethora of work studying efficient algorithms for graph matching under probabilistic models. In this work, we propose a new algorithm for graph matching: Our algorithm associates each vertex with a signature vector using a multistage procedure and then matches a pair of vertices from the two graphs if their signature vectors are close to each other. We show that, for two Erdős–Rényi graphs with edge correlation $1-\alpha$, our algorithm recovers the underlying matching exactly with high probability when $\alpha \le 1 / (\log \log n)^C$, where $n$ is the number of vertices in each graph and $C$ denotes a positive universal constant. This improves the condition $\alpha \le 1 / (\log n)^C$ achieved in previous work.

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 (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). In a sense, our result shows that apart from identification of the weights of parallel 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 asymptotic improvement of the Gilbert-Varshamov bound for 2 < q < 49.

It was conjectured by Levi, Hadwiger, Gohberg and Markus that the boundary of any convex body in $${\mathbb R}^n$$ can be illuminated by at most $$2^n$$ light sources, and, moreover, $$2^n-1$$ light sources suffice unless the body is a parallelotope. We show that if a convex body is close to the cube in the Banach–Mazur metric, and it is not a parallelotope, then indeed $$2^n-1$$ light sources suffice to illuminate its boundary. Equivalently, any convex body sufficiently close to the cube, but not isometric to it, can be covered by $$2^n-1$$ smaller homothetic copies of itself.