# Algebra Seminar

Title: On the Proportion of MRD codes and Semifields

Abstract: Rank-metric codes are subspaces of a full matrix space over a finite field, where we endow the matrix space with the rank metric: d(A,B)=rk(A-B). The rank distance of such a code is defined as the minimum rank of all its nonzero elements. Codes with the maximum possible size for a given rank distance are called MRD codes (maximum rank-distance codes). In this talk I will discuss the proportion of MRD codes within the space of all rank-metric codes of the same dimension. More specifically, I will consider the asymptotic behavior of this proportion as the field size tends to infinity. I will report on the few answers that exist at this point. Special attention will be paid to the case of [3x3;3]-MRD codes, where a close relation to 3-dimensional semifields arises.