摘  要:

Errors correcting codes are a basic primitive which provides robust tools against noise in transmission. On the theoretical perspective, they are usually founded on beautiful properties of some mathematical objects. For example, one of the oldest construction of codes is due to Reed and Solomon and takes advantage of the fact the number of roots of a polynomial cannot exceed its degree. During the last decades, new problems in coding theory have emerged (e.g. secure network transmission or distributive storage) and new families of codes have been proposed. In this perspective, Martínez-Peñas has recently introduced a linearized version of Reed-Solomon codes which, roughly speaking, is obtained by replacing classical polynomials by a noncommutative version of them called Ore polynomials.

In this talk, I will revisit Martínez-Peñas' construction and give a new description of the duals of linearized Reed-Solomon codes. This will lead us to explore the fascinating world of noncommutative polynomials and notably develop a theory of residues for rational differential forms in this context.