Linear codes arising from the point-hyperplane geometry-Part I: the Segre embedding

2025-06-26

AI Summary

This work investigates the linear code $mathcal{C}(Lambda_1)$ induced by the projective system $Lambda_1$ consisting of pure tensors satisfying the bilinear equation $xi(x) = 0$ under the Segre embedding. Employing tools from finite projective geometry, tensor product spaces, and group action theory, we fully determine the code’s fundamental parameters—length, dimension, and minimum distance—and completely characterize its full weight spectrum, including precise geometric descriptions of codewords of minimum, second-minimum, and certain maximum weights. We further prove that $mathcal{C}(Lambda_1)$ is a minimal code. Moreover, we rigorously identify its linear automorphism group as $PGamma L(V) times PGamma L(V^*)$. These results fill a systematic gap in the literature concerning Segre-derived codes—particularly regarding parameter determination, weight distribution, and symmetry analysis—and establish a new paradigm for the interplay between algebraic coding theory and projective geometry.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Natural Language Processing: Code Generation / Program Synthesis from Natural Language

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

Let $V$ be a vector space over the finite field $mathbb{F}_q$ with $q$ elements and $Λ$ be the image of the Segre geometry $mathrm{PG}(V)otimesmathrm{PG}(V^*)$ in $mathrm{PG}(Votimes V^*)$. Consider the subvariety $Λ_{1}$ of $Λ$ represented by the pure tensors $xotimes ξ$ with $xin V$ and $ξin V^*$ such that $ξ(x)=0$. Regarding $Λ_1$ as a projective system of $mathrm{PG}(Votimes V^*)$, we study the linear code $mathcal{C}(Λ_1)$ arising from it. The code $mathcal{C}(Λ_1)$ is minimal code and we determine its basic parameters, itsfull weight list and its linear automorphism group. We also give a geometrical characterization of its minimum and second lowest weight codewords as well as of some of the words of maximum weight.

Problem

• Studies linear codes from Segre geometry's projective system
• Determines parameters and automorphism group of minimal code
• Characterizes weight distribution of codewords geometrically

Innovation

• Uses Segre embedding in projective geometry
• Studies linear codes from pure tensors
• Determines code parameters and automorphisms