Unitals in shift planes of odd order

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 square order. We investigate various geometric and combinatorial properties of these planes, such as the self-duality, the existence of O’Nan configurations, 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).