The proposal focuses on the relation between finite geometry and coding theory. Our attention

goes specifically to maximum rank distance codes or MRD codes. These are a particular type of

rank codes, which describe a systematic way of building codes that could detect and correct

multiple random rank errors.

The aim of this research is to get a deeper understanding of the geometry behind MRD codes, and

more importantly, to further develop the connections between them. For this we consider linear

sets. These are particular sets of points in a finite projective space and are frequently used to

construct or characterise examples of various substructures in finite geometry. The connection

between linear MRD codes and linear sets is fairly new and needs to be developed further. Apart

from classifying known examples, we aim at constructing new examples of MRD codes and their

related linear sets.

As a second goal, we focus on the construction of non-linear MRD codes. There are few general

constructions of non-linear MRD codes known, hence obtaining a new infinite family would be a

big step forward. For this, we consider a different geometric link, namely the link between MRD

codes and exterior sets to a Segre variety. This algebraic variety corresponds to the cartesian

product of two projective spaces and has various connection with field reduction, subgeometries

and linear sets. We will concentrate on constructing non-linear MRD codes by purely geometric

arguments.