## Section: Research Program

### Algebraic Coding Theory

Coding Theory studies originated with the idea of using redundancy in
messages to protect against noise and errors. The last decade of the
20th century has seen the success of so-called iterative decoding
methods, which enable us to get very close to the Shannon
capacity. The capacity of a given channel is the best achievable
transmission *rate* for reliable transmission. The consensus in
the community is that this capacity is more easily reached with these
iterative and probabilistic methods than with algebraic codes (such as
Reed–Solomon codes).

However, algebraic coding is useful in settings other than the Shannon context. Indeed, the Shannon setting is a random case setting, and promises only a vanishing error probability. In contrast, the algebraic Hamming approach is a worst case approach: under combinatorial restrictions on the noise, the noise can be adversarial, with strictly zero errors.

These considerations are renewed by the topic of *list decoding*
after the breakthrough of Guruswami and Sudan at the end of the
nineties. List decoding relaxes the uniqueness requirement of
decoding, allowing a small list of candidates to be returned instead
of a single codeword. List decoding can reach a capacity close
to the Shannon capacity, with zero failure, with small lists, in
the adversarial case.
The method of Guruswami and Sudan enabled list decoding of most of the
main algebraic codes: Reed–Solomon codes and Algebraic–Geometry (AG)
codes and new related constructions “capacity-achieving list
decodable codes”. These results open the way to applications again
adversarial channels, which correspond to worst case settings in
the classical computer science language.

Another avenue of our studies is AG codes over various geometric objects. Although Reed–Solomon codes are the best possible codes for a given alphabet, they are very limited in their length, which cannot exceed the size of the alphabet. AG codes circumvent this limitation, using the theory of algebraic curves over finite fields to construct long codes over a fixed alphabet. The striking result of Tsfasman–Vladut–Zink showed that codes better than random codes can be built this way, for medium to large alphabets. Disregarding the asymptotic aspects and considering only finite length, AG codes can be used either for longer codes with the same alphabet, or for codes with the same length with a smaller alphabet (and thus faster underlying arithmetic).

From a broader point of view, wherever Reed–Solomon codes are used, we can substitute AG codes with some benefits: either beating random constructions, or beating Reed–Solomon codes which are of bounded length for a given alphabet.

Another area of Algebraic Coding Theory with which we are more recently concerned is the one of Locally Decodable Codes. After having been first theoretically introduced, those codes now begin to find practical applications, most notably in cloud-based remote storage systems.