Foundations of additive codes over finite fields

Abstract

Additive codes were initially introduced by Delsarte in 1973 within the context of association schemes and recently they have become of interest due to their application in constructing quantum error-correcting codes. We give foundational results for additive codes where the elements are from a finite field, and define the orthogonality relation using group characters. We introduce a type for these additive codes and explore the notion of independence for a generating set. Additionally, we provide a definition for a generator matrix of an additive code based on its type. We also relate the type of an additive code to the type of its orthogonal. We study a family of kernels and ranks associated with these additive codes. We relate the equivalence of additive codes to their type, the family of kernels and ranks, and duality. We see how these relations contribute in the classification of additive codes. Finally, we provide a general encoding and decoding method for these codes.