  We theoretically studied the electronic structure, magnetic properties, and
lattice dynamics of a series of $3d$ transition-metal antiperovskite compounds
AXM$_{3}$ by density function theory. Based on the Stoner criterion, we drew
the magnetic phase diagram of carbon-based antiperovskites ACM$_{3}$. In the
phase diagram, compounds with non-magnetic ground state but locating near the
ferromagnetic boundary are suggested to yield sizeable electron-phonon coupling
and behave superconductivity. To approve this deduction, we systematically
calculated the phonon spectra and electron-phonon coupling of a series of
Cr-based antiperovskites ACCr$_{3}$ and ANCr$_{3}$. The results show that
AlCCr$_{3}$, GaCCr$_{3}$, and ZnNCr$_{3}$ could be moderate coupling BCS
superconductors. The influence of spin fluctuation on superconductivity are
discussed. Furthermore, other potential superconducting AXM$_{3}$ including
some new Co-base and Fe-based antiperovskite superconductors are predicted from
the magnetic phase diagram.
