The Theory of Cyclic Codes and a Generalization to Additive Codes

We present a new approach to the theory of cyclic and constacyclic codes and generalize the theory to cover the family of additive (not necessarily linear) cyclic codes. The approach is based on the action of the Galois group (cyclotomic cosets). The conventional representation of cyclic codes as ideals in a factor ring of the polynomial ring is not needed. 1 General facts on linear codes The construction of many good linear codes involves the following step: one constructs a code C; which is linear over an extension eld IF q m of IF q : A linear q-ary code is then derived either as the trace code or as the subbeld code of C: Deenition 1. Let F = IF q r jIF q and let C be an F-linear code. Denote the trace by tr = tr : F ?! IF q : The trace code tr(C) is deened as the set of 1