Mini-course «Applications of Global Fields to Error Correcting Codes and Sphere Packings» by Philippe Lebacque (Université de Franche-Comté)

Error correcting codes are at the heart of the digital world and therefore play a central role in modern societies. In the late 70's, V.D. Goppa constructed the first error correcting codes from algebraic geometry. The problem of construction of "good" algebraic geometric codes reduces to the question of finding curves of arbitrary large genus defined over finite fields with many rational points.

After recalling basic definitions and properties concerning codes, we will focus on algebraic geometric codes and on their relations to global fields. We will discuss several constructions of asymptotically good families of curves over finite fields that lead to good codes. Namely,

• explicit equations (after Garcia and Stichtenoth),
• class field theory (after Serre),
• reduction of modular curves (after Ihara and Tsfasman--Vladuts--Zink).

If time permits, we will also cover the constructions of sphere packings (continuous analogues of codes) that use number fields. In the course we will deal with algebraic curves, the Riemann-Roch and Riemann--Hurwitz theorems, the ramification theory, the class field theory, Galois cohomology. All the necessary notions and results will be recalled.

The last lecture will be devoted to the most recent advances concerning invariants of infinite global fields.

Prerequisites.  The audience should be familiar with *basic* algebraic geometry and algebraic number theory, although necessary results might be recalled without proofs.