Refinements of Craig’s lattices and new dense lattice packings

10/30/2012 - 12:15pm
10/30/2012 - 1:10pm
Carmelo Interlando (San Diego State University)

The classical sphere packing problem seeks to find an arrangement of identical spheres in the n-dimensional Euclidean space R^n in such a way that the fraction of the space occupied by them, namely, the density, is the highest possible. Despite the considerable progress that has been achieved towards a solution, the problem remains open in its general form. The problem by itself is an active field of research in contemporary mathematics and has influenced disparate areas of study. Besides its theoretical appeal, a solution is of interest to communication engineers: as demonstrated by the founder of information theory, Claude Shannon, dense packings yield near-optimal signal constellations that can be used for reliable and efficient data transmission.

In this talk, a particular case of the problem, namely, lattice packings, will be discussed. Lattice packings are sphere packings such that the centers of the spheres form a lattice, that is, a discrete subgroup of R^n. Some of our recent contributions, which include the discovery of packings of record density (among the known ones), will be presented. Our main tool is algebraic number theory: a lattice is obtained as the geometrical representation of an ideal from the ring of integers of a number field. The challenges for determining dense lattices are basically: the suitable choices of the field and the ideal in its ring of integers, and the determination of a lower bound on the minimum distance of the associated lattice. Craig’s lattices were discovered in the late 1970's by Maurice Craig; they are the geometrical representations of certain ideals in rings of cyclotomic integers. The construction being presented is a modification of Craig’s method.

Millikan 208 (Pomona College)