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

Let Γ be the point-hyperplane geometry of a projective space PG(V), where V is a $(n+1)$-dimensional vector space over a finite field Fq of order $q.$ Suppose that $σ$ is an automorphism of Fq and consider the projective embedding ε_σ of Γ into the projective space PG(V⊗ V^*) mapping the point ([x],[ξ])∈ Γ to the projective point represented by the pure tensor x^σ⊗ 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 Λ₁=ε₁(Γ). Here we focus on the case σ=1 and we investigate the linear code C(Λ_σ) arising from the projective system Λ_σ=ε_σ(Γ). In particular, after having verified that 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.