Codes and Expansions (CodEx) Seminar

Joey Iverson (Iowa State University)
Equiangular lines over finite fields

A longstanding open problem asks to give the maximum number \(N(d)\) of equiangular lines through the origin in \(\mathbb{C}^d\) or \(\mathbb{R}^d\). In the complex case, it is known that \(N(d) \leq d^2\), and it is widely believed this is attained for EVERY \(d > 1\). In the real case, it is known that \(N(d) \leq d(d+1)/2\), and it is widely believed this is attained for NO \(d > 23\). These are hard problems, and we will not solve them in this talk. What we will do is discuss equiangular lines in classical geometries over finite fields, and show that the analogous bounds are attained infinitely often. We will also discuss interactions with the real and complex problems, and explain how almost every known strongly regular graph arises from equiangular lines over a finite field.

Joint with Gary Greaves, John Jasper, and Dustin Mixon