The theory of linear codes over finite rings has been generalized to linear codes over infinite rings in two special cases; the ring of -adic integers and formal power series ring. These rings are examples of complete discrete valuation rings (CDVRs). In this paper, we generalize the theory of linear codes over the above two rings to linear codes over complete local principal ideal rings. In particular, we obtain the structure of linear and constacyclic codes over CDVRs. For this generalization, first we study linear codes over , where is a commutative Noetherian ring, is a maximal ideal of , and denotes the -adic completion of . We call these codes, -adic codes. Using the structure of -adic codes, we present the structure of linear and constacyclic codes over complete local principal ideal rings.
