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Finite Fourier series and ovals in PG(2, 2h)

Part of: Finite geometry and special incidence structures Designs and configurations

Published online by Cambridge University Press:  09 April 2009

J. Chris Fisher  and
Bernhard Schmidt
J. Chris Fisher
Affiliation:
Department of Mathematics, University of Regina, Regina S4S 0A2, Canada, e-mail: [email protected]
Bernhard Schmidt
Affiliation:
School of Physical and Mathematical Sciences, Nanyang Technological University, No. I Nanyang Walk, B]k 5, Level 3, Singapore, 637616, e-mail: bernhard @ ntu.edu.sg
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Abstract

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We propose the use of finite Fourier series as an alternative means of representing ovals in projective planes of even order. As an example to illustrate the method's potential, we show that the set {wj + w3j + w−3j: 0 ≤ j ≤ 2h} ⊂ GF (22h) forms an oval if w is a primitive (2h + 1)st root of unity in GF(22h) and GF(22h) is viewed as an affine plane over GF(2h). For the verification, we only need some elementary ‘trigonometric identities’ and a basic irreducibility lemma that is of independent interest. Finally, we show that our example is the Payne oval when h is odd, and the Adelaide oval when h is even.

MSC classification

Secondary: 51E20: Combinatorial structures in finite projective spaces 05B25: Finite geometries
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 81 , Issue 1 , August 2006 , pp. 21 - 34
Copyright
Copyright © Australian Mathematical Society 2006

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