On free planes in lattice ball packings

This note, by studying the relations between the length of the shortest lattice vectors and the covering minima of a lattice, proves that for every d-dimensional packing lattice of balls one can find a four-dimensional plane, parallel to a lattice plane, such that the plane meets none of the balls of the packing, provided that the dimension d is large enough. Nevertheless, for certain ball packing lattices, the highest dimension of such 'free planes' is far from d.