Algebraic Semidefinite Programming for Codes and Anti-Codes in Finite Geometry

In the previous years, I have studied various techniques from algebraic combinatorics and their

possible applications to Galois geometry. Algebraic techniques such as eigenvalue methods, linear

programming, clique-coclique bounds, and rank arguments turned out to be very successful in

tackling open problems in finite geometry (upper bounds on subspace codes, EKR theorems). This

is not a one-way street. Galois geometries often provide interesting infinite families of examples

for which techniques from algebraic combinatorics are useful; in turn improving our understanding

of these techniques.

This proposal has the following aims.

Applying new techniques from algebraic combinatorics on important problems within Galois

geometries. Here it was hardly ever tried to apply semidefinite programming, which saw huge

developments during the last 10 years. We will also aim at finding new applications for related

methods such as rank arguments.

Applying semidefinite programming to finite geometries will yield new insights into this technique

itself. On this basis we will aim at developing new techniques in algebraic combinatorics.

There are other areas of combinatorics, where researchers apply similar techniques from algebraic

combinatorics on other structures (permutation groups, matchings, hypergraphs, multisets). To

unify techniques we will establish close contacts with researchers in these areas.