A Family of Asymptotically Good Lattices Having a Lattice in Each Dimension

Abstract

A new constructive family of asymptotically good lattices with respect to sphere packing density is presented. The family has a lattice in every dimension n ≥ 1. Each lattice is obtained from a conveniently chosen integral ideal in a subfield of the cyclotomic field ℚ(ζq) where q is the smallest prime congruent to 1 modulo n.

An Extension of Craig's Family of Lattices

“…, the packing radius of A (i) p−1 is lower bounded by √ pi/2; see [2]. Moreover, for large n = p − 1, these lattice packings satisfy…”

On a generalization of Craig lattices

“…The following result can be compared with the construction of the Craiglike lattices in [3]. Our lattices are obviously denser, since in their construction the prime number q is required to be the smallest prime q satisfying q ≡ 1 mod n. When (> n) is a prime number, the generalized Craig lattice A Proof.…”

“…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]).…”

New lattice sphere packings denser than Mordell-Weil lattices

“…From Theorem 2.1 we have the following analogous Craig lattices for all dimensions with "not bad" densities. The following result can be compared with the construction of Craig-like lattices in [7]. It is obvious our lattices are denser, since in their construction the prime number q is required to be the smallest prime q satisfying q ≡ 1 mod n.…”

“…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]).…”