Abstract

Codes and designs are important objects in combinatorics that are closely connected. Many classical problems—often with applications in areas such as information theory, geometry, and quantum physics—involve finding large codes or small designs. A powerful tool for tackling such problems is the linear programming method, which provides bounds on the size of codes and designs. This method has found applications across different mathematical areas and has led to landmark results, most notably the solution of the sphere packing problem in dimensions 8 and 24. In this talk, we will look at the origins of the linear programming method and explore its applications to several types of codes. We will see how optimal codes are linked to designs, and how probability theory can be used to establish their existence. Finally, we will briefly discuss how these methods lead to new results in finite geometry.