An improvement to the Minkowski‐Hiawka bound for packing superballs

The Minkowski‐Hlawka bound implies that there exist lattice packings of n ‐dimensional “superballs” | x 1 | σ + … + | x n | σ ≤ 1 (σ = 1,2,…) having density Δ satisfying log 2 Δ ≥ − n (l + o (l)) as n → ∞. For each n = p σ ( p an odd prime) we exhibit a finite set of lattices, constructed from codes over GF ( p ), that contain packings of superballs having log 2 Δ ≥ − cn (l + o (l)), where c = 1 + 2 e − 2 π 2log 2   e + … = 1.7719 … for σ = 2 (the classical sphere packing problem), worse than but surprisingly close to the Minkowski‐Hlawka bound, and c = 0·8226 … for σ = 3, c = 0·6742 … for σ = 4, etc., improving on that bound.