Unitals in shift planes of odd order

Abstract

A finite shift plane can be equivalently defined via abelian relative difference sets as well as planar functions. In this paper, we present a generic way to construct unitals in finite shift planes of odd orders $q^2$. We investigate various geometric and combinatorial properties of them, such as the self-duality, the existences of O'Nan configurations, the Wilbrink's conditions, the designs formed by circles and so on. We also show that our unitals are inequivalent to the unitals derived from unitary polarities in the same shift planes. As designs, our unitals are also not isomorphic to the classical unitals (the Hermitian curves).