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

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.