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

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 andWhite and others. In this paper, we consider the questionofwhenGauss periods take exactly three rational values.Weobtain 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 values. Furthermore, we discuss connections between three-valued Gauss periods and combinatorial structures such as circulant weighing matrices and three-class association schemes. Dedicated to Chris Godsil, on the occasion of his 65th birthday T. Feng research supported in part by the Fundamental Research Funds for Central Universities of China and the National Natural Science Foundation of China under Grant 11422112. K. Momihara research supported by JSPS under Grant-in-Aid for Young Scientists (B) 25800093 and Scientific Research (C) 24540013. B Qing Xiang [email protected] Tao Feng [email protected] Koji Momihara [email protected] 1 Department of Mathematics, Zhejiang University, Hangzhou 310027, Zhejiang, People’s Republic of China 2 Faculty of Education, Kumamoto University, 2-40-1 Kurokami, Kumamoto 860-8555, Japan 3 Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA