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, we prove that Grassmann codes, Segre codes, line polar Grassmann codes of orthogonal, symplectic, hermitian type, codes arising from dual polar spaces of orthogonal and symplectic type and codes arising from the point-hyperplane geometry of a projective space are minimal codes.