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.

翻译：设 ${\mathcal C}(Ω)$ 为由射影系统 $Ω$ 在 $\mathrm{PG}(V)$ 上生成的线性码。考虑点线几何 $Γ=({\mathcal P},{\mathcal L})$ 及其投影嵌入 $\varepsilon\colon Γ\rightarrow \mathrm{PG}(V)$。我们证明，若对于 $\mathrm{PG}(V)$ 的任意超平面 $H$，由 $Γ$ 的共线图在集合 $Γ\setminus\varepsilon^{-1}(H)$ 上诱导的图是连通的，则以 $Ω:=\varepsilon(\mathcal{P})$ 作为射影系统得到的射影码为最小码。作为应用，Grassmann 码、Segre 码、正交型、辛型、埃尔米特型的极值 Grassmann 码，以及由射影空间的点-超平面几何产生的码均为最小码。