AbstractThe Minkowski‐Hlawka bound implies that there exist lattice packings ofn‐dimensional “superballs” |x1|σ+ … + |xn|σ≤ 1 (σ = 1,2,…) having density Δ satisfying log2Δ ≥ −n(l +o(l)) asn→ ∞. For eachn=pσ(pan odd prime) we exhibit a finite set of lattices, constructed from codes overGF(p), that contain packings of superballs having log2Δ ≥ −cn(l +o(l)), wherec=1+2e−2π2log2 e+…=1.000000007719…for σ = 2 (the classical sphere packing problem), worse than but surprisingly close to the Minkowski‐Hlawka bound, andc= 0·8226 … for σ = 3,c= 0·6742 … for σ = 4, etc., improving on that bound.