We study lower bounds of the packing density of a system of nonoverlapping equal spheres in R n , n ≥ 2, as a function of the maximal circumradius of its Voronoi cells. Our viewpoint is that of Delone sets which allows to investigate the gap between the upper bounds of Rogers or Kabatjanskiȋ-Levenstein and the Minkowski-Hlawka type lower bounds for the density of lattice-packings, without entering the fundamental problem of constructing Delone sets with Delone constants between 2 −0.401 and 1. As a consequence we provide explicit asymptotic lower bounds of the covering radii (holes) of the BarnesWall, Craig and Mordell-Weil lattices, respectively BW n , A (r) n and M W n , and of the Delone constants of the BCH packings, when n goes to infinity.
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