We will study error correcting codes built using algebraic geometry and number theory, and their applications in cryptography, multi-party computation, and complexity theory. Indeed, many new questions are naturally raised in these application domains, and these questions actually involve deep mathematics.
The problems that we will study in algebraic coding theory are those which have natural applications in the above domains (computer science, public-key cryptography and multi-party computation), and our approach will be to systematic reformulate these questions in terms of algebraic geometry. The standard simple mathematical notions (polynomials, finite fields) will be replaced by their abstract equivalent in algebraic geometry.
We structure the work in three tasks: computing for the multiplicative properties of codes, decoding for new decoding problems, and finally geometry, for studying families of codes buiilt with advanced algebraic geometry.
- The computing task has applications in multi-party computation, algebraic complexity, cryptanalsyis of McEliece’s cryptosystem. The point is to study the properties of codes under component-wise multiplication. The multiplicative properties can also be studied using additive number theory.
- The decoding task deals with non standard questions in decoding: list-decoding, locally decodable codes, subfield subcodes, etc. There has been many breakthroughs in these topics, yet their generalization to higher genus curves or higher dimensional varieties have only been partially explored.
- The geometry task studies new research directions per se in algebraic geometry and coding theory: codes built over higher dimensional varieties, families of codes, rationally connected varieties. This tasks will lead to interesting mathematical basic research.