Three-valued Gauss periods, circulant weighing matrices and association schemes

TL;DR

This paper investigates the conditions under which Gauss periods take exactly three rational values and explores their connections to combinatorial structures like circulant weighing matrices and association schemes.

Contribution

It provides necessary and sufficient conditions for three-valued Gauss periods and links these to combinatorial objects, expanding understanding of their algebraic and combinatorial properties.

Findings

Necessary conditions for three-valued Gauss periods are established.

Numerous examples of three-valued Gauss periods are provided.

Connections to circulant weighing matrices and association schemes are discussed.

Abstract

Gauss periods taking exactly two values are closely related to two-weight irreducible cyclic codes and strongly regular Cayley graphs. They have been extensively studied in the work of Schmidt and White and others. In this paper, we consider the question of when Gauss periods take exactly three rational values. We obtain numerical necessary conditions for Gauss periods to take exactly three rational values. We show that in certain cases, the necessary conditions obtained are also sufficient. We give numerous examples where the Gauss periods take exactly three vlaues. Furthermore, we discuss connections between three-valued Gauss periods and combinatorial structures such as circulant weighing matrices and 3-class association schemes.