Leírás
A Magyar Tudomány Ünnepe, a Fejes Tóth László Díj átadása.
14.15 Előadás: Maryna Viazovska,
15.15 Fejes Tóth László Díj átadása.
Title: Polynomial Freiman-Ruzsa, Reed-Muller codes and Shannon capacity
Abstract: In 1948, Shannon used a probabilistic argument to show the existence of codes achieving a maximal rate defined by the channel capacity. In 1954, Muller and Reed introduced a simple deterministic code construction, based on polynomial evaluations, conjectured shortly thereafter to achieve capacity. The conjecture led to decades of activity involving various areas of mathematics and was finally settled in 2023 by Emmanuel Abbe and Colin Sandorn. In this talk, I will present an alternative proof of the weak form of the capacity result, i.e., that RM codes have a vanishing local error at any rate below capacity. Our proof puts forward a connection with the recent proof of the Polynomial Freiman-Ruzsa conjecture and an entropy extraction approach. Further, a new additive combinatorics conjecture is put forward, with potentially broader applications to coding theory and, in particular, the stronger result of a vanishing global error for RM codes. This is joint work with Emmanuel Abbe, Colin Sandon and Vladyslav Shashkov.