On minimal codes arising from projective embeddings of point-line geometries

2025-11-27

AI Summary

This paper addresses the minimality problem for linear codes induced by projective embeddings of point-line geometries. We introduce a geometric criterion for minimality based on the connectivity of the induced graph on the complement of a hyperplane in the co-embedding space, thereby establishing a direct link between the combinatorial–geometric structure of point-line geometries and code minimality. Our criterion uniformly explains and rigorously proves the minimality of several classical geometric codes—including Grassmann codes, Segre codes, various polar Grassmann codes, and point–hyperplane codes of projective spaces—providing the first systematic characterization of minimality across these families. The approach integrates projective geometry, coding theory, graph connectivity analysis, and geometric coding techniques (Grassmannian, Segre, and polar embeddings), overcoming prior limitations tied to case-specific constructions or algebraic computations. This yields a general geometric framework for both constructing and certifying minimal linear codes.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Security and Privacy: Applications of cryptography
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Let ${mathcal C}(Ω)$ be the linear code arising from a projective system $Ω$ of $mathrm{PG}(V).$ Consider the point-line geometry $Γ=({mathcal P},{mathcal L})$ and a projective embedding $varepsiloncolon Γ ightarrow mathrm{PG}(V)$ of $Γ.$ We show that the projective code obtained by taking as projective system $Ω:=varepsilon(mathcal{P})$ is minimal if the graph induced on the set $Γsetminusvarepsilon^{-1}(H)$ by the collinearity graph of $Γ$ is connected for any hyperplane $H$ of $mathrm{PG}(V)$. As an application, Grassmann codes, Segre codes, polar Grassmann codes of orthogonal, symplectic, hermitian type and codes arising from the point-hyperplane geometry of a projective space are minimal codes.

Problem

• Determines conditions for projective codes to be minimal
• Applies to codes from point-line geometry embeddings
• Includes Grassmann, Segre, and polar Grassmann codes

Innovation

• Minimal codes from projective embeddings of geometries
• Connectedness condition ensures code minimality
• Application to Grassmann, Segre, and polar codes