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 values. Furthermore, we discuss connections between three-valued Gauss periods and combinatorial structures such as circulant weighing matrices and three-class association schemes.
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Acknowledgments
The authors would like to thank both reviewers for their comments and constructive suggestions. In particular, we thank one of the reviewers who gave a short proof of Theorem 2.7, which is the proof presented here in this paper.
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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.
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Feng, T., Momihara, K. & Xiang, Q. Three-valued Gauss periods, circulant weighing matrices and association schemes. J Algebr Comb 43, 851–875 (2016). https://doi.org/10.1007/s10801-016-0664-z
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DOI: https://doi.org/10.1007/s10801-016-0664-z