Abstract: Let F*_{2 ^d} denote a finite field of 2^d elements, with d 2. Examples of (2^d - 1 ,2^d-1 - 1 ,2^d-2 - 1) cyclic difference sets in F*_{2 ^d} include quadratic residue difference sets, GMW difference sets, and difference sets from monomial hyperovals. We show that, except for a few cases with small d, these difference sets are pairwise inequivalent. This is accomplished in part by examining their 2-ranks. The 2-ranks of all these difference sets were previously known, except for those connected with the Segre and Glynn hyperovals. We determine the 2-ranks of the difference sets arising from the Segre and Glynn hyperovals, in the following way. Stickelberger's theorem for Gauss sums is used to reduce the computation of these 2-ranks to a problem of counting certain cyclic binary strings of length d. This counting problem is then solved combinatorially, with the aid of the transfer matrix method. We give further applications of the 2-rank results, including the determination of the nonzeros of certain binary cyclic codes, and a criterion in terms of the trace function to decide for which in F*_{2 ^d} the polynomial x⁶ + x + has a zero in F*_{2 ^d}, when d is odd.

Authors: Ronald Evans, Henk Hollmann, Christian Krattenthaler, and Qing Xiang

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