The problem of maximum packing density of the $n$-dimensional real space with spheres is a classic one. Exact values of the density are known only in a few cases ($n=1,2,3,8,24$), and there have been several recent improvements of the bounds for other small dimensions. In a parallel development, researchers have studied the maximum size of packings of the $n$-dimensional Hamming space, known as error-correcting codes. While existence bounds in both cases are found by random choice, the best known impossibility results are obtained by an application of a general method commonly known as Delsarte’s linear programming. The best known upper bound on the maximum size of a code with a given minimum distance for large $n$ was obtained in 1977, and it has proved surprisingly resistant to various improvement attempts, including the semidefinite programming extension of LP.

In the first part of the talk we will introduce the general problem and give an overview of the known results on upper bounds on codes and related problems such as equiangular lines and families of finite sets with restricted intersections. In the second part, we will delve into the details of the proofs for the case of codes and highlight some obstacles for further improvements.