We study quadratic residue difference sets, GMW difference sets, and difference sets arising from monomial hyperovals, all of which are (2(d) - 1, 2(d-1) - 1, 2(d-2) - 1) cyclic difference sets in the multiplicative g...
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We study quadratic residue difference sets, GMW difference sets, and difference sets arising from monomial hyperovals, all of which are (2(d) - 1, 2(d-1) - 1, 2(d-2) - 1) cyclic difference sets in the multiplicative group of the finite field F-2d of 2(d) elements, with d greater than or equal to 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 cyclicbinarystrings 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 binarycyclic codes, and a criterion in terms of the trace function to decide for which beta in F-2d* the polynomial x(6) + x + beta has a zero in F-2d, when d is odd. (C) 1999 Academic Press.
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