Half-metallic antiferromagnets are the ideal materials for spintronic applications since their zero magnetization leads to lower stray fields and thus tiny energy losses. Starting from the ${\mathrm{Mn}}_{2}\mathrm{V}\mathrm{Al}$ and ${\mathrm{Mn}}_{2}\mathrm{V}\mathrm{Si}$ alloys we substitute Co and Fe for Mn and we show by means of first-principles electronic structure calculations that the resulting compounds are ferrimagnets. When the total number of valence electrons reaches the magic number of 24 the Fe-doped compounds are semimetals and thus nonmagnetic while the Co-doped ones show the desirable half-metallic antiferromagnetic character. The compounds are very likely to be synthesized experimentally since the parent compounds ${\mathrm{Mn}}_{2}\mathrm{V}\mathrm{Al}$ and ${\mathrm{Co}}_{2}\mathrm{V}\mathrm{Al}$ have been already grown in the Heusler $L{2}_{1}$ lattice structure.