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

2025-07-22

AI Summary

This paper investigates linear codes induced by twisted embeddings of point-hyperplane geometries over finite fields via nontrivial field automorphisms. Using projective systems as a framework, the authors combine techniques from pure tensor representations in tensor products, group action analysis, and geometric combinatorics to systematically characterize the coding-theoretic properties of these codes. Key contributions include: proving minimality; determining all parameters—length, dimension, and minimum distance—exactly; providing precise geometric characterizations of codewords of minimum and second-minimum weight; establishing the maximum weight when both $q$ and $n$ are odd; and fully describing the structure of the automorphism group. This work fills a theoretical gap in the study of linear codes constructed via twisted embeddings and extends the methodological scope of projective geometric coding theory.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Security and Privacy: Applications of cryptography
• Systems and Infrastructure for Web, Mobile and WoT: Location- and context-aware Web and WoT applications and services

Abstract

Let $arΓ$ 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 $arΓ$ into the projective space $mathrm{PG}(Votimes V^*)$ mapping the point $([x],[ξ])in arΓ$ to the projective point represented by the pure tensor $x^σotimes ξ$, 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 $Λ_1=varepsilon_{1}(arΓ).$ Here we focus on the case $σ ot=1$ and we investigate the linear code ${mathcal C}(Λ_σ)$ arising from the projective system $Λ_σ=varepsilon_σ(arΓ).$ 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.

Problem

• Investigates linear codes from twisted point-hyperplane embeddings
• Determines parameters and automorphism group of minimal codes
• Characterizes minimum and second lowest weight codewords geometrically

Innovation

• Twisted embedding for projective geometry
• Minimal code with determined parameters
• Geometrical characterization of codewords