Complex spherical designs and codes

TL;DR

This paper introduces the concept of complex spherical designs, demonstrating that many possess association scheme structures, which are typically symmetric in real designs, thus expanding the understanding of complex spherical configurations.

Contribution

The paper develops the notion of complex spherical designs and shows that many of them form nonsymmetric association schemes, a novel insight compared to real designs.

Findings

Many complex spherical designs carry association scheme structures

These association schemes are nonsymmetric, unlike real spherical designs

The work extends the theory of designs to the complex sphere context

Abstract

Real spherical designs and real and complex projective designs have been shown by Delsarte, Goethals, and Seidel to give rise to association schemes when the strength of the design is high compared to its degree as a code. In contrast, designs on the complex unit sphere remain relatively uninvestigated, despite their importance in numerous applications. In this paper we develop the notion of a complex spherical design and show how many such designs carry the structure of an association scheme. In contrast with the real spherical designs and the real and complex projective designs, these association schemes are nonsymmetric.