Codes closed under arbitrary abelian group of permutations

Algebraic structure of codes closed under arbitrary abelian group G of permutations is investigated resulting in insight into dual of G-invariant codes and self-dual G-invariant codes. For special types of the groups, these codes give cyclic, abelian, quasi-cyclic and quasi-abelian codes. Karlin's decoding algorithm for systematic one-generator quasicyclic codes is extended for systematic quasi-abelian codes with any number of generators.