Gauss Sums, Jacobi Sums, and $p$-ranks of Cyclic Difference Sets

Details Gauss Sums, Jacobi Sums, and $p$-ranks of Cyclic Difference Sets Gauss Sums, Jacobi Sums, and $p$-ranks of Cyclic Difference Sets

1998-07-07

arxiv.org/abs/math/9807029v2

J. Combin. Theory Ser. A 87 (1999), 174-219

Mathematics

Combinatorics

Proofs of the main theorems 4.6 and 4.8 significantly simplified; now only 37 pages, AmS-LaTeX; to appear in J. Combin. Theory

Scientific paper

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 $F_{2^d}$ of $2^d$ elements, with $d \geq 2$. 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 $F_{2^d}^*$ the polynomial $x^6 + x + \beta$ has a zero in $F_{2^d}$, when $d$ is odd.

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