On a class of quasi-Hermitian surfaces in even characteristic

TL;DR

This paper classifies a special class of quasi-Hermitian varieties in even characteristic projective spaces, analyzes their line structures in three dimensions, determines their automorphism groups, and relates them to minimal codes.

Contribution

It provides a classification of quasi-Hermitian varieties in arbitrary dimensions and characterizes their automorphisms and code equivalences in the three-dimensional case.

Findings

Complete classification of quasi-Hermitian varieties up to projective equivalence.

Determination of the automorphism group of $ ext{H}_ ext{ε}^3$.

Proof of equivalence of certain minimal codes.

Abstract

In [1], a new quasi-Hermitian variety $\mathcal{H}_\varepsilon^r$ in $\mathrm{PG}(r, q^2)$, with $q = 2^e$ and $e \geq 3$ an odd integer, was constructed. The variety depends on a primitive element $\varepsilon$ of the underlying field $\mathrm{GF}(q^2)$.11 In the present paper, we first provide a classification of such varieties up to projective equivalence in finite projective spaces of arbitrary dimension. Then, we focus on the case $r = 3$ and study the structure of the lines contained in $\mathcal{H}_\varepsilon^3$; as a consequence, we determine the full automorphism group of $\mathcal{H}_\varepsilon^3$ . Finally, as a byproduct, we prove the equivalence of certain minimal codes introduced in [3].