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Properties of the Leech Lattice Properties of the Leech Lattice Austin Roberts University of Puget Sound April 14, 2006 Abstract This paper catalogues and describes the properties of the Leech lattice and gives a basic introduction to the fields where it emerges. Topics include sphere packing, error- correcting codes, and simple groups, along with some related topics such as kissing num- bers and designs. The focus is primarily geometric and combinatorial. The discussion is aimed towards an advanced undergraduate student though it is understood to be more meaningful with some prior exposure to the covered topics. 1 1 Introduction This paper catalogues and describes the properties of the Leech lattice and gives a basic intro- duction to the fields where it emerges. Topics include sphere packing, error-correcting codes, and simple groups, along with some related topics such as kissing numbers and designs. The discussion is aimed towards an advanced undergraduate student, though it is understood to be more meaningful with some prior knowledge of the covered topics. The Leech lattice is a remarkable mathematical object. For all of the time and thought that went in to its creation, it is a surprisingly simple geometric object, that happens to be of profound importance in mathematics. The topic is made all the more interesting with an understanding of how recently much of it has come about. The discovery of the Leech lattice was heavily dependent on error-correcting codes, which were not established until the late 1940’s, and by the late 1970’s J. H. Conway had found three new simple groups within the automorphism group of the lattice. This paper also includes work as recent as 2004. It is difficult to encompass all of this work in a narrative fashion. There is no simple choice of where to begin because each topic is dependent on another for proof techniques and explanations. This paper chooses to begin by presenting the Leech lattice in one of its purest forms, as a particularly nice generator matrix. Though at first unmotivated, all of the necessary information about the Leech lattice lies in this 24×24 matrix, and so we take the liberty of investigating it before understanding how it was discovered. Lastly, it should be remembered that almost all fields discussed are the topics of ongoing work. All problems associated with sphere packing have proven very difficult, as have many of those associated with coding. Simple groups, although totally enumerated, continue to be the source for much ongoing work in group theory. We are in the middle of exciting work. The results contained here are milestones, but also stepping stones. We are thus discussing matters of importance and wide application that are all the richer for their relevance. 2 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 2 2 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 2 2 0 0 0 0 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 2 2 0 0 2 2 0 0 2 2 0 0 2 2 0 0 0 0 0 0 0 0 0 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 0 0 0 0 0 0 0 0 2 0 0 2 2 0 0 2 2 0 0 2 2 0 0 2 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 2 0 2 0 2 0 0 2 2 2 0 0 0 0 0 0 2 2 0 0 0 0 0 0 2 0 0 2 2 0 2 0 2 2 0 0 2 0 2 0 2 0 2 0 0 0 0 0 2 2 0 0 2 0 2 0 2 0 0 2 0 0 0 0 2 0 0 2 0 0 0 0 0 2 2 2 2 0 0 0 2 0 0 0 2 0 0 0 2 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 2 2 0 0 2 2 0 0 2 2 0 0 2 2 0 0 0 0 0 0 0 0 0 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 −3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 A matrix representation of the Leech lattice (lines included for ease of viewing). 2 Sphere Packing, Kissing, and Covering In this section we introduce the context in which the Leech lattice was first discovered and describe some of the lattice’s basic properties. Some calculations, such as the proof of the Leech lattice’s kissing number, are postponed until codes and designs are discussed in Section 3. 2.1 The Sphere Packing Problem When Leech first discovered his lattice, he was searching for a solution to the sphere packing problem in 24 dimensions. The problem dates back to Thomas Harriot in the late 1590’s, more famously to Johaness Kepler in the beginning of the 17th century, and eventually to a review that Gauss wrote in 1831. It asks, is there a denser arrangement of equally sized spheres than the standard way of stacking oranges? This arrangement, known as the face-centered cubic lattice (fcc lattice), was not proven to be optimal until 1998 (see [14] for the current proof); 3 however, the heavy dependence of this proof on computers is still controversial. The modern version of the problem is generalized to n dimensions. The goal of sphere packing is to find the packing, in a given number dimensions, which achieves the highest possible density ∆, defined as the proportion of space taken up by spheres. Though the formal definition of density is rarely used, we provide it here along with a definition of radius. Definition 1 (Radius) Given a set of points P in IRn, the packing radius of P is 1 ρ = min({ 2 · dist(pi, pj) | pi, pj ∈ P, i 6= j}) where dist is the standard Euclidean distance between two points. It is important to notice that the packing radius, and in fact, all other properties are entirely determined by the sphere centers. We will assume that ρ is positive and will often use the term arrangement, instead of set, to imply that ρ 6= 0. Definition 2 (Density) Given a arrangement of points P in IRn with packing radius ρ, the packing density of P is n ! ave (vol({x ∈ IR | dist(x, pi) ≤ ρ}) ∆(P ) = pi ∈ P n vol({x ∈ IR | dist(x, pi) ≤ dist(x, pj) for all j 6= i}) where ave is the average over a set and vol is the n-dimensional volume. In practice, finding the volume of the broad sets in the definition of ∆ is often bypassed by taking advantage of symmetries within an arrangement rather than considering all of n- dimensional space. The process of arranging spheres in n dimensions is less than intuitive, but we may begin to understand by looking in lower dimensions. The sphere packing problem takes place in Euclidean space, IRn and so we define a sphere as the set of points within a given distance from a central point. In the 1-dimensional case a 1-sphere is merely a line segment within the Euclidean space consisting of a single line (see Fig.1). Figure 1: The optimal packing in one dimension If we give our spheres a radius of one unit and place the center of the spheres on the even integers of a number line marking off the distances of E1, then it is easy to see that we can cover the entire space, achieving 100% density. The scale of this arrangement, as with all arrangements, is strictly a matter of convenience and causes no difference to the density or other traits that we will examine.
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