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

Abstract

Let $\barΓ$ 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 $σ$ is an automorphism of $\mathbb{F}_q$ and consider the projective embedding $\varepsilon_σ$ of $\barΓ$ into the projective space $\mathrm{PG}(V\otimes V^*)$ mapping the point $([x],[ξ])\in \barΓ$ to the projective point represented by the pure tensor $x^σ\otimes ξ$, with $ξ(x)=0.$ 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 $σ=1$ and we studied the projective code arising from the projective system $Λ_1=\varepsilon_{1}(\barΓ).$ Here we focus on the case $σ\not=1$ and we investigate the linear code ${\mathcal C}(Λ_σ)$ arising from the projective system $Λ_σ=\varepsilon_σ(\barΓ).$ In particular, after having verified that $\mathcal{C}( Λ_σ)$ 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.