Classifying additive codes.

摘要

We introduce vector-valued character theory, a higher dimensional generalization of character theory. We then use vector-valued characters to extend the MacWilliams equivalence theorem for linear codes to a class of additive codes, called admissible Galois-embedded additive codes. We show that the class of linear codes is an admissible class of Galois-embedded additive codes and we offer a method for constructing large classes of non-linear admissible Galois-embedded additive codes. Subsequently, we identify methods for determining the equivalence of additive codes over F9 via ternary linear codes. Finally, we generalize the Hamming, split Hamming and the complete weight enumerator polynomials to the generalized m-split weight enumerators and identify MacWilliams-type identities relating the generalized m-split weight enumerator polynomials of codes to those of their duals.