The Minkowski‐Hlawka bound implies that there exist lattice packings of n‐dimensional “superballs” |x1|σ + … + |xn|σ ≤ 1 (σ = 1,2,…) having density Δ satisfying log2 Δ ≥ −n(l + o(l)) as n → ∞. For each n = pσ (p an odd prime) we exhibit a finite set of lattices, constructed from codes over GF(p), that contain packings of superballs having log2 Δ ≥ −cn(l + o(l)), where c=1+2e−2π2log2 e+…=1.000000007719… 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.
Summary. If of ___ ~" is a bounded, convex body which is symmetric through each of the coordinate hyperplanes, then there exist codes which give rise, via Construction A of Leech and Sloane, to lattice-packings of off whose density A satisfies the logarithmic Minkowski-Hlawka bound, lim inf,_~ ~logzx ~ > -1. This follows as a corollary of our main result, Theorem 9, a general way of obtaining lower bounds on the lattice-packing densities of various bodies. Unfortunately, when n is at all large, it is computationally prohibitive (although theoretically possible) to exhibit the arrangements explicitly.
The n‐dimensional cross polytope, |x|+|x2|+…+|xn≤1, can be lattice packed with density δ satisfying
ln δ⩾−n/2+o(n) as n→∞,
but proofs of this, such as the Minkowski‐Hlawka theorem, do not actually provide such packings. That is, they are nonconstructive. Here we exhibit lattice packings whose density satisfies only
ln δ⩾−n ln ln n+O(n),
but by a highly constructive method. These are the densest constructive lattice packings of cross polytopes obtained so far.
We obtain explicit lower bounds on the lattice packing densities δL of superballs G of quite a general nature, and we conjecture that as the dimension n approaches infinity, the bounds are asymptotically exact. If the conjecture were true, it would follow that the maximum lattice‐packing density of the Iσ‐ball |x1|σ+…+|xn|σ⩽1 is 2−n(1+σ(1)) for each σ in the interval 1 ≤ σ ≤ 2.
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