Finite geometry and its partners in crime: codes, graphs, and equiangular lines.

Finite geometry has made significant contributions to multiple areas of combinatorics, such as coding
theory, extremal combinatorics, and graph theory. The goal of this project is to advance this exciting
interplay in the following three work packages.
1. MDS codes are some of the most desirable objects in coding theory. However, long MDS codes
seem to be quite rare. Their scarcity has been confirmed for the class of linear codes. The goal of the
project is to understand whether a similar phenomenon occurs in the wider class of additive MDS
codes over finite fields, or whether there are new families of long MDS codes to be discovered.
2. A set of lines in a real or complex vector space is called equiangular if any two lines have the
same angle. Determining the maximum number of equiangular lines is a central question in
combinatorics and quantum information theory. Recently, the finite field version of this problem
gained traction. We will further investigate equiangular lines over finite fields and the interplay
between the finite and infinite settings.
3. Inversive spaces are a remarkable family of incidence geometries, which have received a
surprisingly small amount of attention so far. We will investigate the algebraic structure of graphs
that naturally arise from the blocks of inversive spaces.
This project will make me a more versatile researcher, and increase my capability to make significant
contributions in multiple areas of combinatorics, using finite geometry.