On the 2-ranks of a class of unitals

TL;DR

This paper studies the binary codes derived from certain unitals in shift planes, providing new lower bounds on their dimensions using Kloosterman sums, which improve previous results especially for fields of size 3^m.

Contribution

It introduces improved lower bounds on the dimensions of codes from unitals in shift planes, utilizing Kloosterman sums for the first time in this context.

Findings

New lower bounds on code dimensions for unitals in shift planes.

Bounds are tighter than previous results by Leung and Xiang.

Explicit formulas for bounds when q=3^m, depending on parity of m.

Abstract

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 $m$.