α-Flocks with Oval Herds and Monomial Hyperovals

Abstract

In PG (3, q), q even, Cherowitzo made a detailed study of flocks of a cone with a translation oval as base; also called -flocks [4]. To a flock of a quadratic cone in PG(3, q), q even, there always corresponds a set of q#1 ovals in PG(2, q), called an oval herd. To an -flock of a cone with an arbitrary translation oval as base, there corresponds a herd of q#1 permutation polynomials. For some, but not for all, known examples of -flocks, these q#1 permutation polynomials define an oval herd. This leads to the fundamental problem of determining which -flocks correspond to an oval herd. This article studies a class of -flocks and explicitly describes which members of this class have an associated oval herd. To achieve this goal, all monomial hyperovals +(1, t, tI)"" t3F O ,6+(0, 1, 0), (0, 0, 1), with k"2G#2H, iOj, are determined.1998 Academic Press

On the classification of hyperovals

“…Suppose, for a contradiction, that Φ f is not absolutely irreducible. Let φ j be defined by (3). Our proof relies on the following claim.…”

“…In order to prove Theorem 1.2, we first use the constraints given by Lemmas 2.1, 2.2, and 2.3 and then show that in all remaining cases, Φ f has an absolutely irreducible factor over F q unless f is one of the polynomials in Theorem 1.2. To do so, we frequently use the polynomials (3) φ j (x, y, z) = x(y j + z j ) + y(x j + z j ) + z(x j + y j ) (x + y)(x + z)(y + z) .…”

Arithmetic of Finite Fields

“…Any mapping F on F 2 m that satisfies (5) is called an o-polynomial [7]. Using (3), every o-polynomial results in a bent function in class H. We list below the known o-polynomials due to Frobenius, Segre (1962), Glynn (1983), Cherowitzo (1998) and Payne (1985). The remaining two known cases (Subiaco and Adelaide) are listed in [7].…”

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

“…This conjecture still remains open. Some progress was made in a recent paper by Cherowitzo and Storme [7]. Lemma 2.3 [8,22].…”