On Quasi-cyclic Codes over Integer Residue Rings

  • Maheshanand
  • Siri Krishan Wasan
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4851)


In this paper we consider some properties of quasi-cyclic codes over the integer residue rings. A quasi-cyclic code over ℤ k , the ring of integers modulo k, reduces to a direct product of quasi-cyclic codes over \({\mathbb{Z}}_{p_i^{e_i}}\), \(k = \prod_{i=1}^s p_i^{e_i}\), p i a prime. Let T be the standard shift operator. A linear code \(\mathcal{C}\) over a ring R is called an l-quasi-cyclic code if \(T^l(c) \in \mathcal{C}\), whenever \( c\in \mathcal{C}\). It is shown that if (m, q) = 1, q = p r , p a prime, then an l-quasi-cyclic code of length lm over ℤ q is a direct product of quasi-cylcic codes over some Galois extension rings of ℤ q . We have discussed about the structure of the generator of a 1-generator l-quasi-cyclic code of length lm over ℤ q . A method to obtain quasi-cyclic codes over ℤ q , which are free modules over ℤ q , has been discussed.


Quasi-cyclic codes circulant matrices Galois rings Hensel’s lift 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Maheshanand
    • 1
  • Siri Krishan Wasan
    • 2
  1. 1.Centre for Development of Advanced Computing, NoidaIndia
  2. 2.Department of Mathematics, Jamia Millia Islamia, New DelhiIndia

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