# A characterization of Hermitian varieties as codewords

## TL;DR

This paper characterizes quasi-Hermitian varieties in projective spaces as codewords, extending known results about Hermitian varieties and unital structures in finite geometry.

## Contribution

It generalizes the characterization of Hermitian varieties as codewords to quasi-Hermitian varieties in higher-dimensional projective spaces.

## Key findings

- Quasi-Hermitian varieties are characterized as codewords in certain projective spaces.
- Extension of Blokhuis, Brouwer, and Wilbrink's result to higher dimensions.
- Applicable to projective spaces over prime and prime power fields.

## Abstract

It is known that the Hermitian varieties are codewords in the code defined by the points and hyperplanes of the projective spaces $PG(r,q^2)$. In finite geometry, also quasi-Hermitian varieties are defined. These are sets of points of $PG(r,q^2)$ of the same size as a non-singular Hermitian variety of $PG(r,q^2)$, having the same intersection sizes with the hyperplanes of $PG(r,q^2)$. In the planar case, this reduces to the definition of a unital. A famous result of Blokhuis, Brouwer, and Wilbrink states that every unital in the code of the points and lines of $PG(2,q^2)$ is a Hermitian curve. We prove a similar result for the quasi-Hermitian varieties in $PG(3,q^2)$, $q=p^{h}$, as well as in $PG(r,q^2)$, $q=p$ prime, or $q=p^2$, $p$ prime, and $r\geq 4$.