Cyclic R-additive codes

Bierbrauer (2012) developed the theory of q-linear cyclic codes over (Fq)m and he obtained a parametric description of such codes by cyclotomic cosets. Recently, Cao et al. (2015) obtained the structure of cyclic additive codes over the Galois ring GR(pℓ,m), where m is a prime integer. Let R be a finite commutative ring and Rn=R[x]∕〈xn−1〉. In this paper, we generalize the theory of Fq-linear codes over vector spaces to R-linear codes over free R-algebras (free as R-module). We call these codes, R-additive codes. We introduce a one-to-one correspondence between the classes of cyclic R-additive code and the classes of Rn-linear code. Using the structure of Rn-linear codes, we present the structure of cyclic R-additive codes, where R is a chain ring. Among other results, q-linear cyclic codes over (Fq)m are described by ring-theoretic facts, and the structure of cyclic additive codes over the Galois ring GR(pℓ,m) is given for an arbitrary integer m, not necessarily a prime number.