Acta mathematica scientia,Series B ›› 2013, Vol. 33 ›› Issue (2): 413-422.doi: 10.1016/S0252-9602(13)60008-9

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ON THE CHARACTERIZATION OF CYCLIC CODES OVER TWO CLASSES OF RINGS

 LIU Xiu-Sheng   

  1. School of Mathematics and Physics, Hubei Polytechnic University, Huangshi 435003, China
  • Received:2011-04-08 Revised:2012-07-08 Online:2013-03-20 Published:2013-03-20
  • Supported by:

    The author is supported by the Natural Science Foundation of Hubei Province (B20114410) and the Natural Science Foundation of Hubei Polytechnic University (12xjz14A).

Abstract:

Let R be a finite chain ring with maximal ideal (γ) and residue field F, and let γ be of nilpotency index t. To every code C of length n over R, a tower of codes C = (Cγ0) ⊆ (Cγ) ⊆ … ⊆ (Cγi) ⊆ … ⊆ (C : γt−1) can be associated with C, where for any rR, (C : r) = {eRn | reC}. Using generator elements of the projection of such a tower of codes to the residue field F, we characterize cyclic codes over R. This
characterization turns the condition for codes over R to be cyclic into one for codes over the residue field F. Furthermore, we obtain a characterization of cyclic codes over the formal power series ring of a finite chain ring.

Key words: Finite chain rings, formal power series rings, cyclic codes, tower of codes, Hensel lift

CLC Number: 

  • 94B05
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