Семинар НУЛ АГП "Error-correcting codes from toric varieties"

When storing or transmitting data, errors may accumulate. To address this, one adds redundant information to a message, and this process is formalized by the concept of error-correcting codes. Codes obtained from evaluating rational functions on algebraic varieties tend to have good parameters, which can, however, be notoriously hard to compute. One therefore turns to toric geometry, in which many algebro-geometric computations reduce to combinatorial problems on fans and polytopes.

First, we will discuss the concept of error-correcting codes, their parameters and some classical bounds and constructions; we will then proceed to describe the construction of toric codes and study known bounds on their parameters.