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Cyclic Codes over Galois Rings

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Algorithms and Discrete Applied Mathematics (CALDAM 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9602))

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Abstract

Let R be a Galois ring of characteristic \(p^a\), where p is a prime and a is a natural number. In this paper cyclic codes of arbitrary length n over R have been studied. The generators for such codes in terms of minimal degree polynomials of certain subsets of codes have been obtained. We prove that a cyclic code of arbitrary length n over R is generated by at most \(min\{a, t+1\}\) elements, where \(t=max\{deg(g(x))\},\) g(x) a generator. In particular, it follows that a cyclic code of arbitrary length n over finite fields is generated by a single element. Moreover, the explicit set of generators so obtained turns out to be a minimal strong Gröbner basis.

J. Kaur — Work submitted in partial fulfillment of requirements for the degree of Doctor of Philosophy.

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References

  1. Abualrub, T., Oehmke, R.: Cyclic codes of length \(2^e\) over \(\mathbb{Z}_4.\). Discrete Appl. Math. 128(1), 3–9 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  2. Adams, W., Loustaunau, P.: An Introduction to Gröbner Basis. American Mathematical Society, Providence (1994)

    MATH  Google Scholar 

  3. Byrne, E., Fitzpatrick, P.: Gröbner bases over Galois rings with an application to decoding alternant codes. J. Symbolic Comput. 31, 565–584 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  4. Calderbank, A.R., Sloane, N.J.A.: Modular and \(p\)-adic cyclic codes. Des. Codes Cryptogr. 6(1), 21–35 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  5. Dinh, H.Q., Lopez-Permouth, S.R.: Cyclic and negacyclic codes over finitechain rings. IEEE Trans. Inform. Theory 50(8), 1728–1744 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  6. Dougherty, S.T., Ling, S.: Cyclic codes over \(\mathbb{Z}_4\) of even length. Des. Codes Cryptogr. 39(2), 127–153 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. Dougherty, S.T., Park, Y.H.: On modular cyclic codes. Finite Fields Appl. 13, 31–57 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. Garg, A., Dutt, S.: Cyclic codes of length \(2^k\) over \(\mathbb{Z}_{2^m}.\). Int. J. Eng. Res. Dev. 1(9), 34–37 (2012)

    Google Scholar 

  9. Kanwar, P., Lopez-Permouth, S.R.: Cyclic codes over the integers modulo \(p^m\). Finite Fields Appl. 3(4), 334–352 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  10. Kiah, H.M., Leung, K.H., Ling, S.: Cyclic codes over \(GR(p^2, m)\) of length \(p^k\). Finite Fields Appl. 14(3), 834–846 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Lopez-Permouth, S.R., Ozadam, H., Ozbudak, F., Szabo, S.: Polycyclic codesover Galois rings with applications to repeated-root constacyclic codes. Finite Fields Appl. 19(1), 16–38 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. McDonald, B.R.: Finite Rings with Identity. Marcel Dekker, New York (1974)

    MATH  Google Scholar 

  13. Norton, G.H., Sălăgean, A.: Strong Gröbner bases for polynomials over a principal ideal ring. Bull. Aust. Math. Soc. 64(3), 505–528 (2001)

    Article  MATH  Google Scholar 

  14. Norton, G.H., Sălăgean, A.: Cyclic codes and minimal strong Gröbner bases over a principal ideal ring. Finite Fields Appl. 9(2), 237–249 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  15. Rajan, B.S., Siddiqi, M.U.: Transform domain characterization of cyclic codes over \(\mathbb{Z}_{m}.\). Appl. Algebra Eng. Commun. Comput. 5(5), 261–275 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  16. Sălăgean, A.: Repeated-root cyclic and negacyclic codes over a finite chain ring. Discrete Appl. Math. 154(2), 413–419 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  17. Sobhani, R., Esmaeili, M.: Cyclic and negacyclic codes over the Galois ring \(GR(p^2, m).\). Discrete Appl. Math. 157(13), 2892–2903 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Wan, Z.X.: Finite fields and Galois rings. World Scientific Publishing Company, Singapore (2011)

    Book  Google Scholar 

  19. Woo, S.S.: Ideals of \({\mathbb{Z}_{p^n}}[x]/\langle x^l-1 \rangle.\). Commun. Korean Math. Soc. 26(3), 427–443 (2011)

    Article  MathSciNet  Google Scholar 

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Acknowledgments

The author (Jasbir Kaur) gratefully acknowledges the World Bank funded TEQIP-II for financial support.

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Authors and Affiliations

  1. Department of Applied Sciences, PEC University of Technology, Chandigarh, India

    Jasbir Kaur, Sucheta Dutt & Ranjeet Sehmi

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  1. Jasbir Kaur
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  2. Sucheta Dutt
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  3. Ranjeet Sehmi
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Correspondence to Ranjeet Sehmi .

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Editors and Affiliations

  1. Indian Institute of Science, Bangalore, India

    Sathish Govindarajan

  2. Carleton University, Ottawa, ON, Canada

    Anil Maheshwari

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Kaur, J., Dutt, S., Sehmi, R. (2016). Cyclic Codes over Galois Rings. In: Govindarajan, S., Maheshwari, A. (eds) Algorithms and Discrete Applied Mathematics. CALDAM 2016. Lecture Notes in Computer Science(), vol 9602. Springer, Cham. https://doi.org/10.1007/978-3-319-29221-2_20

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  • DOI: https://doi.org/10.1007/978-3-319-29221-2_20

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-29220-5

  • Online ISBN: 978-3-319-29221-2

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