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.
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