This material has been published in J. Combin. Theory Ser. A 87 (1999), 74-119, the only definitive repository of the content that has been certified and accepted after peer review. Copyright and all rights therein are retained by Academic Press. This material may not be copied or reposted without explicit permission.

Ronald Evans, Henk Hollmann, Christian Krattenthaler and Qing Xiang

Gauss sums, Jacobi sums, and p-ranks of cyclic difference sets

(37 pages)

Abstract. 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 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. 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 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 \beta in F2d* the polynomial x6+x+\beta has a zero in F2d, when d is odd.
See the supplement to the paper.

The following versions are available:

Back to Christian Krattenthaler's home page.