Density doubling, double-circulants, and new sphere packings

Abstract

Abstract. New nonlattice sphere packings in dimensions 20, 22, and 44-47 that are denser than the best previously known sphere packings were recently discovered. We extend these results, showing that the density of many sphere packings in dimensions just below a power of 2 can be doubled using orthogonal binary codes. This produces new dense sphere packings in R n for n = 25, 26, . . . , 31 and 55, 56, . . . , 63. For n = 27, 28, 29, 30 the resulting packings are denser than any packing previously known.

New lattice sphere packings denser than Mordell-Weil lattices

“…From [22] if there was a binary linear [60, 27,16] code, then a new denser lattice with center density 2 18.1039 could be constructed. The Elkies lattice in dimension 60 has center density 2 19.04 , which is the section of Mordell-Weil lattice(see [48]). If there was a binary linear [60, 28,16] code( [22]), we could construct a lattice sphere packing in dimension 60 with center density 2 19.1039 .…”

“…Since [84, 16,32], [85,16,32] and [87,17,32] linear binary codes exist( [22]), we have better lattices in dimension 84 with center density at least 2 35.4 , better lattices in dimension 85 with center density at least 2 35.83 , and better lattice in dimension 86 with center density at least 2 37.33 In the following table 3 we list some possible better lattices under the condition that some nice codes exist. The Elkies lattices of dimensions 57 − 60 are cross-sections of Mordell-Weil lattices, we refer to [48], page 278.…”

“…Kepler conjecture about the densest 3 dimensional sphere packing problem was proved in [26]. Many known densest sphere packings are lattice packings or packings from finitely many translates of lattices(see [11,12,31,47,48]). Constructing lattice sphere packings from error-correcting codes, algebraic number fields and algebraic geometry have been proposed by many authors and stimulated many further works( [30,31,11,15,16,2,36,37,38,33,34,23,24,17,18,19,7,20,42,43,35,47,48,40,41]).…”

“…Many known densest sphere packings are lattice packings or packings from finitely many translates of lattices(see [11,12,31,47,48]). Constructing lattice sphere packings from error-correcting codes, algebraic number fields and algebraic geometry have been proposed by many authors and stimulated many further works( [30,31,11,15,16,2,36,37,38,33,34,23,24,17,18,19,7,20,42,43,35,47,48,40,41]). Recently Leech lattice, which was found in 1965 in [30], has been proved to be the unique densest lattice packing in dimension 24 (see [9,10]).…”

“…This situation was changed in 1995 and 1997, when A. Vardy published [22] and R. Bacher published [1]. New non-lattice sphere packings better than laminated lattice packings were constructed in Euclid spaces of dimensions 20(see [47]), 22(see [13]), 27,28,29, 30(see [48]) and 18(see [4]). Lattice sphere packings in Euclid spaces of dimensions 27, 28 and 29 which are better than laminated lattice packings were constructed in [1].…”