Skew codes

We generalize the notion of cyclic code and we construct codes as ideals in finite quotients of non-commutative polynomial rings, so called Ore rings. We propose a method to obtain block codes of prescribed rank or Hamming distance. In particular we construct a $[42,14,21]_8$ code by imposing a rank and a $[40,20,10]_4$ code by imposing a distance, which both improve by one the minimum distance of the previously best known linear codes with the same length and dimension over those fields. We also study some multivariate Ore rings and the generalization of the Buchberger's algorithm allows us to manipulate the ideals of these rings and to build skew codes. We obtain, in particular, the generator matrix in the standard form.