On a refinement of Craig’s lattices

Abstract

a b s t r a c tLet p be an odd prime. A family of (p − 1)-dimensional over-lattices yielding new record packings for several values of p in the interval [149 . . . 3001] is presented. The result is obtained by modifying Craig's construction and considering conveniently chosen Z-submodules of Q(ζ ), where ζ is a primitive pth root of unity. For p ≥ 59, it is shown that the center density of the (p − 1)-dimensional lattice in the new family is at least twice the center density of the (p − 1)-dimensional Craig lattice.

On a generalization of Craig lattices

“…Since n ≥ 1222, we have n ) in the range 1298 ≤ n ≤ 3482. Thus the lattices above are denser than those in [15]. Some of them are even denser than any previously known lattice.…”

“…A cyclotomic construction of the Leech lattice was given by Craig in [11]. In [15] a refinement of Craig lattices was proposed by adding some fractional numbers. The center densities of the refined Craig lattices are at least three times that of the original Craig lattices.…”

“…The center densities of the refined Craig lattices are at least three times that of the original Craig lattices. These provided some new records; see [15].…”

“…from error-correcting codes, algebraic number fields and algebraic varieties have been proposed by many authors and stimulated many further works ( [22,23,6,10,11,25,12,13,14,3,15,28]). Recently the Leech lattice, which was found in 1965 in [22], has been proved to be the unique densest lattice packing of dimension 24 (see [4,5]).…”

“…For example, the densest known 4096-dimensional sphere packing is a Mordell-Weil lattice with center density 2 11527 . For dimensions in the range 149 ≤ n = p − 1 ≤ 3001 (where p is a prime number, 149 ≤ n ≤ 3001 except p = 509, 513 and 521), many of the known n = (p − 1)-dimensional densest sphere packings are Craig's lattices and their recent refinement (see [10,11,15,6]). In the range 4100 ≤ n ≤ 12754608 or n ≤ 8 · 10 8 , the best known sphere packings are the lattices given by the Bos-Conway-Sloane construction ( [2], [6], page 17, Table 1.3 and Chapter 8, Section 10).…”