The fundamental question of coding theory

A good code has both rate k/n and relative minimum distance d/n large. The fundamental question of coding theory is to describe the closure of the set of points (d/n,k/n). Goppa’s conjecture says that, except for isolated points, it’s the region below the curve y=1-H(x) for x<1/2, where H is the binary entropy function. I'll describe this and a tantalizing connection with hyperelliptic curves that led us briefly to think we'd disproved the conjecture. (Joint work with Jing Hao.)