On the 2-ranks of a class of unitals

Let $U_\theta$ be a unital defined in a shift plane of odd order $q^2$, which are constructed recently by the authors. In particular, when the shift plane is desarguesian, $U_\theta$ is a special Buekenhout-Metz unital formed by a union of ovals. We investigate the dimensions of the binary codes derived from $U_\theta$. By using Kloosterman sums, we obtain a new lower bound on the aforementioned dimensions which improves the result obtained by Leung and Xiang in 2009. In particular, for $q=3^m$, this new lower bound equals $\frac{2}{3}(q^3+q^2-2q)-1$ for even $m$ and $\frac{2}{3}(q^3+q^2+q)-1$ for odd $m$.Comment: arXiv admin note: text overlap with arXiv:1508.0727

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 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)