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

TL;DR

This paper investigates conditions under which projective codes derived from point-line geometries are minimal, and applies these results to various well-known classes of codes.

Contribution

It establishes a criterion for minimality of codes from projective embeddings and demonstrates minimality for several important classes of codes.

Findings

The code is minimal if the induced graph on the geometry minus a hyperplane is connected.

Grassmann, Segre, and polar Grassmann codes are minimal under certain conditions.

Codes from point-hyperplane geometries of projective spaces are minimal.

Abstract

Let ${\mathcal C}(\Omega)$ be the linear code arising from a projective system $\Omega$ of $\mathrm{PG}(V).$ Consider the point-line geometry $\Gamma=({\mathcal P},{\mathcal L})$ and a projective embedding $\varepsilon\colon \Gamma\rightarrow \mathrm{PG}(V)$ of $\Gamma.$ We show that the projective code obtained by taking as projective system $\Omega:=\varepsilon(\mathcal{P})$ is minimal if the graph induced on the set $\Gamma\setminus\varepsilon^{-1}(H)$ by the collinearity graph of $\Gamma$ 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.