turkmath.org

Rank-metric codes are linear spaces of matrices over a finite field, in which the rank of any non-zero matrix is bounded from below by a given integer. MRD codes are those rank-metric codes that meet the rank-metric analogue of the Singleton bound (i.e., those having the largest dimension for the given correction capability). They can be viewed as the q-analogue of MDS codes, and they share with them several structural properties. This talk focuses on the open problem of determining the asymptotic density of MRD codes within the set of codes sharing the same dimension, both as the field size and the matrix column length go to infinity. I will give an overview of the problem and of the various techniques that have been used to (partially) solve it, including recent developments based on graph theory and on the theory of semifields.