Lattice packing of nearly-euclidean balls in spaces of even dimension

Rush, J. A.

doi:10.1017/s0013091500022872

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2022

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“…Take logarithm and we obtain inequality (22). For inequality (23), we use equation (20) and inequality (22). So…”

Section: (mentioning

confidence: 99%

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On the lower bound for packing densities of superballs in high dimensions

Xie¹,

Ge²

2022

Preprint

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Define the superball with radius r and center 0 in R n to be the setwhich is a generalization of ℓ p -balls. We give two new proofs for the celebrated result that for 1 < p ≤ 2, the translative packing density of superballs in R n is Ω(n/2 n ). This bound was first obtained by Schmidt, with subsequent constant factor improvement by Rogers and Schmidt, respectively. Our first proof is based on the hard superball model, and the second proof is based on the independence number of a graph. We also investigate the entropy of packings, which measures how plentiful such packings are.

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“…Take logarithm and we obtain inequality (22). For inequality (23), we use equation (20) and inequality (22). So…”

Section: (mentioning

confidence: 99%

“…Moreover, Elkies et al [6] improved the Minkowski-Hlawka bound exponentially for superballs and reals p > 2, and they also obtained lower bound for the packing density of more general bodies. For the lower bound constructed by error correcting codes, see Rush [20,21,22] and Liu and Xing [14].…”

Section: Introductionmentioning

confidence: 99%