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An improvement to the Minkowski-Hiawka bound for packing superballs

Published online by Cambridge University Press:  26 February 2010

Jason A. Rush  and
N. J. A. Sloane
Jason A. Rush
Affiliation:
Department of Mathematics, University of California at San Diego, La Jolla, CA 92093, U.S.A.
N. J. A. Sloane
Affiliation:
Mathematical Sciences Research Center, AT & T Bell Laboratories, Murray Hill, NJ07974, U.S.A.
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Abstract

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 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.

Type
Research Article
Information
Mathematika , Volume 34 , Issue 1 , June 1987 , pp. 8 - 18
Copyright
Copyright © University College London 1987

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