Linear codes arising from the point-hyperplane geometry -- Part II: the twisted embedding

Abstract

Let $\bar{\Gamma}$ be the point-hyperplane geometry of a projective space $\mathrm{PG(V)},$ where $V$ is a $(n+1)$-dimensional vector space over a finite field $\mathbb{F}_q$ of order $q.$ Suppose that $\sigma$ is an automorphism of $\mathbb{F}_q$ and consider the projective embedding $\varepsilon_{\sigma}$ of $\bar{\Gamma}$ into the projective space $\mathrm{PG}(V\otimes V^*)$ mapping the point $([x],[\xi])\in \bar{\Gamma}$ to the projective point represented by the pure tensor $x^{\sigma}\otimes \xi$, with $\xi(x)=0.$ In [I. Cardinali, L. Giuzzi, Linear codes arising from the point-hyperplane geometry -- part I: the Segre embedding (Jun. 2025). arXiv:2506.21309, doi:10.48550/ARXIV.2506.21309] we focused on the case $\sigma=1$ and we studied the projective code arising from the projective system $\Lambda_1=\varepsilon_{1}(\bar{\Gamma}).$ Here we focus on the case $\sigma\not=1$ and we investigate the linear code ${\mathcal C}(\Lambda_{\sigma})$ arising from the projective system $\Lambda_{\sigma}=\varepsilon_{\sigma}(\bar{\Gamma}).$ In particular, after having verified that $\mathcal{C}( \Lambda_{\sigma})$ is a minimal code, we determine its parameters, its minimum distance as well as its automorphism group. We also give a (geometrical) characterization of its minimum and second lowest weight codewords and determine its maximum weight when $q$ and $n$ are both odd.