Rational power series, sequential codes and periodicity of sequences |
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Authors: | Xiang-Dong Hou Sergio R López-Permouth |
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Institution: | a Department of Mathematics, University of South Florida, Tampa, FL 33620, United States b Department of Mathematics, Ohio University, Athens, OH 45701, United States |
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Abstract: | Let R be a commutative ring. A power series f∈Rx]] with (eventually) periodic coefficients is rational. We show that the converse holds if and only if R is an integral extension over Zm for some positive integer m. Let F be a field. We prove the equivalence between two versions of rationality in Fx1,…,xn]]. We extend Kronecker’s criterion for rationality in Fx]] to Fx1,…,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 F is both left and right sequential (bisequential). We prove that the converse holds if and only if F is algebraic over Fp for some prime p. Finally, we show that all sequential codes are obtained by a simple and explicit construction. |
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Keywords: | 13F25 94B15 |
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