J. Aust. Math. Soc.  81 (2006), 21-34
Finite Fourier series and ovals in PG$(2,2^h)$

J. Chris Fisher
  Department of Mathematics
  University of Regina
  Regina S4S 0A2
  Canada
  fisher@math.uregina.ca
and
Bernhard Schmidt
  School of Physical and Mathematical Sciences
  Nanyang Technological University
  No. 1 Nanyang Walk, Blk 5, Level 3
  Singapore 637616
  bernhard@ntu.edu.sg


Abstract
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 $\{w^j+w^{3j}+w^{-3j}: 0\le j\le 2^h\}\subset {\rm GF} (2^{2h})$ forms an oval if $w$ is a primitive $(2^h+1)^{\rm st}$ root of unity in ${\rm GF} (2^{2h})$ and ${\rm GF} (2^{2h})$ is viewed as an affine plane over ${\rm GF} (2^h)$. 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.
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