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

Let $\mathcal{C}(\mathrm{\Omega })$ be the linear code arising from a projective system $\mathrm{\Omega }$ of $\mathrm{P}\mathrm{G}(V)\mathrm{.}$ Consider the point-line geometry $\mathrm{\Gamma }=(\mathcal{P},\mathcal{L})$ and a projective embedding $\epsilon \text{\hspace{0.17em}⁣}:\mathrm{\Gamma }\to \mathrm{P}\mathrm{G}(V)$ of $\mathrm{\Gamma }\mathrm{.}$ We show that the projective code obtained by taking as projective system $\mathrm{\Omega }:=\epsilon (\mathcal{P})$ is minimal if the graph induced on the set $\mathrm{\Gamma }\setminus {\epsilon }^{-1}(H)$ by the collinearity graph of $\mathrm{\Gamma }$ is connected for any hyperplane $H$ of $\mathrm{P}\mathrm{G}(V)$. 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.