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A lower bound on packing density. (English) Zbl 0659.10033
If $$H\subseteq {\mathbb{R}}^ n$$ 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 H whose density $$\Delta$$ satisfies the logarithmic Minkowski-Hlawka bound, $$\liminf_{n\to \infty}\log_ 2^ n\sqrt{\Delta}\geq -1.$$ 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.
Reviewer: J.A.Rush

MSC:
 11H31 Lattice packing and covering (number-theoretic aspects) 52C17 Packing and covering in $$n$$ dimensions (aspects of discrete geometry) 11H06 Lattices and convex bodies (number-theoretic aspects)
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References:
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