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

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.