Unitals in shift planes of odd order

TL;DR

This paper introduces a new generic construction of unitals in finite shift planes of odd order, exploring their properties and demonstrating their inequivalence to classical and previously known unitals.

Contribution

It provides a novel, generic method to construct unitals in shift planes of odd order and analyzes their geometric and combinatorial properties.

Findings

Constructed new unitals in shift planes of odd order

Proved these unitals are inequivalent to classical unitals

Analyzed properties like self-duality and O'Nan configurations

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).