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

TL;DR

The paper develops a geometric criterion for the minimality of projective codes arising from embeddings of point-line geometries, framing minimality in terms of cutting blocking sets and maximal hyperplanes. It proves that a connectivity condition on the collinearity graph after removing hyperplane preimages ensures minimality, and it shows that several important families—Grassmann, Segre, and polar Grassmann codes, including symplectic, orthogonal, and hermitian variants—satisfy this condition. Consequently, these codes are minimal, with explicit parameters and characterizations of minimum-weight codewords provided for each family. The results unify geometric and coding-theoretic perspectives and yield minimal codes with large automorphism groups and clear combinatorial structures, with implications for efficient decoding and secret-sharing applications.

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 $\varepsilon\colon Γ\rightarrow \mathrm{PG}(V)$ 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 $Γ\setminus\varepsilon^{-1}(H)$ 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.

Theorems & Definitions (34)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Proposition 2.1
• Definition 2.2
• Definition 2.3
• Proposition 2.4
• Proposition 3.1
• proof
• Remark 1