Codes as Modules over Skew Polynomial Rings

Abstract

In previous works we considered codes defined as ideals of quotients of skew polynomial rings, so called Ore rings of automorphism type. In this paper we consider codes defined as modules over skew polynomial rings, removing therefore some of the constraints on the length of the skew codes defined as ideals. The notion of BCH codes can be extended to this new approach and the skew codes whose duals are also defined as modules can be characterized. We conjecture that self-dual skew codes defined as modules must be constacyclic and prove this conjecture for the Hermitian scalar product and under some assumptions for the Euclidean scalar product. We found new [56, 28, 15], [60,30,16], [62,31,17], [66,33,17] Euclidean self-dual skew codes and new [50,25,14], [58,29,16] Hermitian self-dual skew codes over F ₄, improving the best known distances for self-dual codes of these lengths over F ₄.