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

TL;DR

This paper investigates the properties of a specific class of linear codes derived from a twisted projective embedding of point-hyperplane geometries over finite fields, focusing on their parameters, automorphisms, and weight distributions.

Contribution

It extends previous work by analyzing the case where the automorphism is non-trivial, establishing the code's minimality, parameters, automorphism group, and characterizing key codewords.

Findings

The code is minimal.

Parameters and minimum distance are explicitly determined.

Automorphism group and weight distribution are characterized.

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.