Static magnetic configurations of thin, circular, soft (permalloy) magnetic nanodots, coupled to a hard antidot matrix with perpendicular magnetization, are studied by micromagnetic simulations. It is demonstrated that dipolar fields of the antidot matrix promote the formation of a magnetic-vortex state in nanodots. The vortex is the dot ground state at zero external field in ultrathin nanodots with diameters as low as 60 nm, which is far beyond the vortex stability range in an isolated permalloy nanodot. Depending on the geometry and antidot-matrix material, it is possible to stabilize either a radial vortex state or unconventional vortices with the angle between in-plane magnetization and radial direction $\psi \ne 0,\pi /2$.