Heide Gluesing-Luerssen (Kentucky)

will speak on

On Sparseness of MRD codes and Semifields

Time: 2:00PM
Date: Thu 12th September 2019
Location: Seminar Room SCN 1.25 [map]

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Abstract: Rank-metric codes are subspaces of a full matrix space over a finite field, and 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. The question arises as to how large a rank-metric code can be for a given rank distance. Codes with the maximum size 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. It turns out that, for instance, [3x2;2]-MRD codes have an asymptotic proportion of 1/3. On the other hand, for [3x3;3]-MRD codes this asymptotic proportion is zero, and therefore we call this class of MRD codes sparse. The proof of the sparsity follows from a parametrization of 3-dimensional semifields by Menichetti (1973). I will discuss the relation between MRD codes and semifields and present the main steps of the proof.

(This talk is part of the Algebra and Number Theory series.)

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