Giacomo Micheli (Oxford)
Permutations and codes using a density method

Abstract: The construction of permutation functions over a finite field is a task of great interest in applied areas such as cryptography and coding theory. In this talk I describe a method which combines Chebotarev density theorem with elementary group theory to produce permutation rational functions over a finite field. The method is entirely constructive and as a corollary we get the classification of permutation polynomials up to degree 4 over any finite field of odd characteristic. Time permitting, I briefly explain what is a locally recoverable code (LRC) and describe another application of a similar method which allows to build optimal LRCs by solving a Galois theoretical problem.