Groebner bases, Geometric codes and Order Domains

Course description

Algebraic Geometry codes enjoy a rich mathematical structure and have excellent properties, but they are difficult to explicitly encode/decode. Also, accurate estimates for their distance are difficult. Recently, Order Domain codes have been proposed to partially bypass these limitations by exploiting a pure Groebner basis approach with explicit code constructions.

We will explain their properties, show how to compute their parameters and how to decode them, with emphasis on Groebner basis theoretical tools. We will describe their relation to other codes, such as one-point geometric Goppa codes and affine-variety codes. We will provide all the necessary commutative algebra background, including Groebner basis theory. Some tutorials with programmes running on new computational facilities will be given to help the students experiment with the methods exposed in the course. Lecture notes covering the whole course will be given to all participants.