An improvement to the Minkowski-Hlawka bound for packing superballs.

The Minkowski-Hlawka bound implies that there exist lattice packings of n-dimensional "superballs" |chi sub 1| sup sigma + ... +| chi sub n | sup sigma = - n(1 + o(1)) as n -> infinity. For each n = p sup sigma (p an odd prime) we exhibit a finite set of lattices, constructed from codes over GF(p), that contain packings of superballs having log sub 2 delta >= - c n(1+o(1)), where c = 1 + 2e sup (-2pi) sup 2 log sub 2 e + ... = 1.000000007719... for sigma = 2 (the classical sphere packing problem), worse than but surprisingly close to the Minkowski-Hlawka bound, and c = 0.8226... for sigma = 3, c = 0.6742... for sigma = 4, etc., improving on that bound.