Real equiangular lines and related codes

Abstract

We consider real equiangular lines and related codes. The driving question is to find the maximum number of equiangular lines in a given dimension. In the real case, this is controlled by combinatorial phenomena, and until only very recently, the exact number has been unknown. The complex case appears to be driven by other phenomena, and configurations are conjectured always to meet the absolute bound of d^2 lines in dimension d. We consider a variety of the techniques that have been used to approach the problem, both for constructing large sets of equiangular lines, and for finding tighter upper bounds. Many of the best-known upper bounds for codes are instances of a general linear programming bound, which we discuss in detail. At various points throughout the thesis, we note applications in quantum information theory.