Tittel: Constructing the Leech lattice through the extended Golay code
Veileder: Øystein Skarsæterhagen
Sammendrag: Imagine we want to send information from a source to a destination, but on its way something slightly alters the information. Small alterations change the interpretation of the information completely. Because of this, mathematicians and others have developed techniques for correcting small alterations the information has obtained in transmission. These techniques are known as error correcting codes. In this thesis we look at a few important examples of codes and how to construct them. We further use these codes to construct what is known as sphere packings.
Imagine a crate filled with balls. Knowing that the balls are equal in size, what is the configuration of balls that allows us to fit the most balls in a crate? This is the origin of the densest sphere packing problem. Instead of looking at the amount of balls, we can look at the density, in other words the amount of space occupied by the balls. Now imagine that instead of a crate, we have all of $\mathbb{R}^3$. What is the the densest possible packing here? The generalized sphere packing problem asks this question for $\mathbb{R}^n$ for all $n$. Intuitively, the densest packing problem is to find sphere centers such that spheres of equal size occupy the most space without any spheres overlapping.
Surprisingly the codes and sphere packings are connected for some special cases; such as the Leech Lattice, the best packing in 24 dimensions. Historically the error correcting codes were the foundation needed to find some of the sphere packings, and this is why we will focus on codes first to later construct the Leech lattice from one of most important error correcting codes. More on the history behind the error correcting codes and the connection with sphere packings can be found in [11].