An alternative construction of B-M and B-T unitals in Desarguesian planes

TL;DR

This paper introduces a novel method to construct non-classical B-M and B-T unitals in Desarguesian planes by redefining the geometric model with specific lines and algebraic curves.

Contribution

It provides a new construction approach for non-classical B-M and B-T unitals using a non-standard projective plane model with particular geometric elements.

Findings

Constructed non-classical B-M unitals in a new geometric model.

Extended the construction to B-T unitals with algebraic curves.

Demonstrated the viability of alternative models for unital construction.

Abstract

We present a new construction of non-classical unitals from a classical unital $U$ in $PG(2,q^2)$. The resulting non-classical unitals are B-M unitals. The idea is to find a non-standard model $\Pi$ of $PG(2,q^2)$ with the following three properties: 1. points of $\Pi$ are those of $PG(2,q^2)$; 2. lines of $\Pi$ are certain lines and conics of $PG(2,q^2)$; 3. the points in $U$ form a non-classical B-M unital in $\Pi$. Our construction also works for the B-T unital, provided that conics are replaced by certain algebraic curves of higher degree.