Rational power series, sequential codes and periodicity of sequences

Let RR be a commutative ring. A power series f∈R[[x]]f∈R[[x]] with (eventually) periodic coefficients is rational. We show that the converse holds if and only if RR is an integral extension over ZmZm for some positive integer mm. Let FF be a field. We prove the equivalence between two versions of rationality in F[[x1,…,xn]]F[[x1,…,xn]]. We extend Kronecker’s criterion for rationality in F[[x]]F[[x]] to F[[x1,…,xn]]F[[x1,…,xn]]. We introduce the notion of sequential code which is a natural generalization of cyclic and even constacyclic codes over a (not necessarily finite) field. A truncation of a cyclic code over FF is both left and right sequential (bisequential). We prove that the converse holds if and only if FF is algebraic over FpFp for some prime pp. Finally, we show that all sequential codes are obtained by a simple and explicit construction.