Gauss Sums, Jacobi Sums, and p-Ranks of Cyclic Difference Sets

We study quadratic residue difference sets, GMW difference sets, and difference sets arising from monomial hyperovals, all of which are (2d−1, 2d−1−1, 2d−2−1) cyclic difference sets in the multiplicative group of the finite field F2d of 2d elements, with d⩾2. We show that, except for a few cases with small d, these difference sets are all pairwise inequivalent. This is accomplished in part by examining their 2-ranks. The 2-ranks of all of 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. Stickelbergerʹs 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 formulas, including the determination of the nonzeros of certain binary cyclic codes, and a criterion in terms of the trace function to decide for which β in F*2d the polynomial x6+x+β has a zero in F2d, when d is odd.