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

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:https://doi.org/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.