Pure Gauss sums and skew Hadamard difference sets

McEliece (1974), Evans (1981) and Aoki (1997, 2004, 2012) studied Gauss sums, some integral powers of which are in the field of rational numbers. Such Gauss sums are called {\it pure}. In particular, Aoki~(1997) classified pure Gauss sums for extension degrees $f=1,2,3,4$. In this paper, as a continuous study, we further characterize pure Gauss sums for odd extension degrees and classify them for $f=5,7,9,11,13,17,19,23$. Furthermore, we characterize a special subclass of pure Gauss sums in view of an application for skew Hadamard difference sets. Based on the result, we give a new construction of skew Hadamard difference sets, which is a large generalization of that given by Feng and Xiang (2013).

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