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

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