Further lattice packings in high dimensions - CiteSeerX

MATHEMATIKA A JOURNAL OF PURE AND APPLIED MATHEMATICS VOL. 29. PART 2.

December 1982.

No. 58

FURTHER LATTICE PACKINGS IN HIGH DIMENSIONS A. BOS, J. H. CONWAY AND N. J. A. SLOANE

Abstract. Barnes and Sloane recently described a "general construction" for lattice packings of equal spheres in Euclidean space. In the present paper we simplify and further generalize their construction, and make it suitable for iteration. As a result we obtain lattice packings in Um with density A satisfying log 2 A ~ — m logf m, as m -> oo, where logf m is the smallest value of k for which the fc-th iterated logarithm of m is less than 1. These appear to be the densest lattices that have been explicitly constructed in high-dimensional space. New records are also established in a number of lower dimensions, beginning in dimension 96. §1. Introduction. In the past 23 years a series of papers [1-3, 5, 8-12, 15-17] have described a variety of methods for packing equal spheres in Euclidean space. The present paper continues the sequence by simplifying the "general construction" of Barnes and Sloane [1], and thereby eliminating the requirement that the initial lattice be generated by its minimal vectors. This enables us to iterate the construction, obtaining lattice packings in Um with density A satisfying log 2 A ~ — m log* m ,

as m -> oo ,

(1)

where logf m is the smallest value of k for which the fc-th iterated logarithm of m is less than 1. These appear to be the densest lattice packings that have been explicitly constructed in high-dimensional space. Non-lattice packings with log2A > — 6m + o(m) were constructed in [15] (and non-lattice packings satisfying (1) in [3]), but there is still room for improvement since the best lattice or non-lattice packings are known to lie in the range — m< log2 A < —0-599m+ o(m),

as m -* oo

(2)

(see [4, 7, 14, 18]). The new construction also generalizes that given in [1] so as to use codes over other alphabets. The construction is described in Section 2, and Section 3 contains a number of applications. A selection of the best packings known in dimensions up to 2 2 0 will be found in Table 1. [MATHEMATIKA, 29 (1982), 171-180]

172

A. BOS, J. H. CONWAY AND N. J. A. SLOANE

Notation. The norm of a vector x is its squared length x . x. If Lm is a lattice in Rm (usually the subscript indicates the dimension), its minimum norm M is min {x . x : x e L m , x =£ 0}, its determinant detL m is the volume of a fundamental region, and its density A and centre density S are given by

where Vm is the volume of an m-dimensional sphere of unit radius. In Section 3 we also use the parameters H = jm, y = log 2 5 .

All the packings mentioned in this paper are lattice packings.

§2. The Construction. The ingredients for the construction are a lattice A in Um, an endomorphism D of A that satisfies certain conditions, one of which is that A/DA is an elementary abelian group E of order pb, say, and a family of codes C o 2 Cj 2 ... 2 Ca of length n over E; the result is a family L o £ L t s ... s La of lattice packings in Rm".

(i) Let A be a lattice in W with minimum non-zero norm M. (ii) Let D be an endomorphism of A which is also a similarity {i.e. a constant times an orthogonal transformation) and which satisfies

pD-1 = £ afD;

(3)

i=0

for integers p ^ 1, r > 0, a 0 ,..., a r . Let T = D~l, so that, from (3), pT is also an endomorphism of A. (iii) Assume that A/DA has the structure of an elementary abelian group E of order pb for some integer b ^ 1. This implies that there is a b-dimensional sublattice K £ A, spanned, say, by vectors vx,..., vb e A, such that K/(DA n K) s £ . The assumptions also imply that pK £ K £ A ,

(4)

pKepAeDAEA,

(5)

and therefore that K/pK = K/(DA nK)=*

E.

(6)

Note that (4) and (5) imply pK £ DA n K, so K/pX 2 K/(DA n K). But

FURTHER LATTICE PACKINGS

173

both sides have order pb and so must be equal, and (6) follows. Furthermore |det T| = \ = r, say ,

(7)

and T multiplies norms by t2. (iv) Assume that all pb — 1 non-zero congruence classes of TA/A have minimum norm at least t2M. (v) Let (p denote the natural map from Z/pZ to Z which takes the congruence class x to x, for x e {0, 1,..., p — 1}. The elements of E may be identified with the b-tuples X = (xl,..., xb), where all xt e Z/pZ. Then X ^ V{X) = codewords and has minimum distance dt (we indicate this by saying that Ct has parameters [n, k(, d;]), and suppose that Co is the trivial [n,n, 1] code. Let c l5 ..., cbka e £" be chosen so that a typical codeword of C; can be written as bk,

X XyCj, X;eZ/pZ, for j = 1,..., a. The New Lattices. Let Lo = A" and, for i = 1, 2,..., a, define L- =

11