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 _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=₁ (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=_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.