Two new sequences of ovals in finite desarguesian planes of even order

Glynn, David G.

doi:10.1007/bfb0071521

Cited by 51 publications

(60 citation statements)

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“…By Corollary (2) to Theorem 1.8 there are q + 2 o-polynomials equivalent to / , and the proof is complete. (2,4). PROOF.…”

Section: Q) Q = 2 H > 4 Are the Following: (1) The Regular Hyperovamentioning

confidence: 96%

“…By Theorem 1.3 both / and g are squares, say / = f 2 and g = g\. Now /,(a) 2 = g { {a) 2 for all a e GF(q), so f x {a) = g x (a) for all a £ GF(q). Therefore /j = g x (mod* 9 +x); so f 2 = g 2 (mod* 2 " + x 2 ) .…”

Section: If F Is An O-polynomial For Pg(2q R ) With Coefficients In mentioning

confidence: 97%

“…For example, q = 8 and q = 32 have q -1 prime so hyperovals in PG (2,8) or PG{2, 32) can never have a binomial o-polynomial.…”

Section: Proof Suppose That F(x) -Axmentioning

confidence: 99%

“…In PG (2,16) In fact the stabiliser of the Lunelli-Sce hyperoval was first found in [11], without using Hall's result. This was used, together with Theorems 1.5 and 1.8 to prove Hall's result without a computer [7].…”

Section: O-polynomials Formentioning

confidence: 99%

“…The hyperovals in PG (2,32) have not yet been classified, and as a consequence the number of o-polynomials for PG(2, 32) is unknown.…”

Section: O-polynomials For Pg{2 32)mentioning

confidence: 99%

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Polynomials for hyperovals of Desarguesian planes

O'Keefe

Penttila

1991

J Aust Math Soc A

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This paper studies o-polynomials, that is, polynomials which represent hyperovals in Desarguesian projective planes of even order. We present theoretical restrictions on the form that o-polynomials can have, and we determine the number of o-polynomials corresponding to each of the known classes of hyperovals (other than Cherowitzo's). We use this to give the number of known o-polynomials for the fields of orders 4, 8, 16 and 32. Exploratory computer searches for o-polynomials for fields of small orders greater than 16 are reported.1991 Mathematics subject classification (Amer. Math. Soc.) 51 E 20. Theoretical resultsLet PG(2, q) be the projective plane over the field GF{q) of order q, where q is a power of a prime p . A hyperoval of PG{2, q) is a set of q + 2 points, no three of which are collinear. An account of hyperovals appears in [5], but here we note a few relevant definitions and results. Firstly, hyperovals exist in PG(2, q) if and only if q is even. A hyperoval is regular if it contains q + 1 points which are the points of a non-degenerate conic in PG (2, q). Conversely, the points of a non-degenerate conic together with a unique further point called the nucleus of the conic provide an example of a hyperoval. THEOREM 1.1 [5, 8.4.1]. If q = 2 then every hyperoval is regular.

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“…By Corollary (2) to Theorem 1.8 there are q + 2 o-polynomials equivalent to / , and the proof is complete. (2,4). PROOF.…”

Section: Q) Q = 2 H > 4 Are the Following: (1) The Regular Hyperovamentioning

confidence: 96%

Section: If F Is An O-polynomial For Pg(2q R ) With Coefficients In mentioning

confidence: 97%

“…For example, q = 8 and q = 32 have q -1 prime so hyperovals in PG (2,8) or PG{2, 32) can never have a binomial o-polynomial.…”

Section: Proof Suppose That F(x) -Axmentioning

confidence: 99%

Section: O-polynomials Formentioning

confidence: 99%

“…The hyperovals in PG (2,32) have not yet been classified, and as a consequence the number of o-polynomials for PG(2, 32) is unknown.…”

Section: O-polynomials For Pg{2 32)mentioning

confidence: 99%

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Polynomials for hyperovals of Desarguesian planes

O'Keefe

Penttila

1991

J Aust Math Soc A

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α-Flocks with Oval Herds and Monomial Hyperovals

Cherowitzo¹,

Storme²

1998

Finite Fields and Their Applications

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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

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Decompositions of Difference Sets

Jungnickel

Tonchev

1999

Journal of Algebra

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We characterize those symmetric designs with a Singer group G which admit a quasi-regular G-invariant partition into strongly induced symmetric subdesigns. In terms of the corresponding difference sets, the set associated with the larger design can be decomposed into a difference set describing the small designs and a suitable relative difference set. This generalizes the decomposition of the classical design Ž . with the complements of hyperplanes in PG m y 1, q as blocks into sub-designs Ž . arising from PG d y 1, q whenever d divides m. Parametrically, these geometrical examples provide the only known examples of the situation we are studying. But there are many nonisomorphic examples with the same parameters, namely the complements of the classical GMW designs and some generalizations. We also discuss the possibilities for obtaining new difference sets in this way and point out a connection to the recent constructions of Ionin for symmetric designs. ᮊ 1999 Academic Press

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Two new sequences of ovals in finite desarguesian planes of even order

Abstract: According to the definition of the famous

Cited by 51 publications

References 5 publications

Polynomials for hyperovals of Desarguesian planes

Polynomials for hyperovals of Desarguesian planes

α-Flocks with Oval Herds and Monomial Hyperovals

Decompositions of Difference Sets

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