On a class of quasi-Hermitian surfaces in even characteristic

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].