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On quasi-cyclic codes over \({\mathbb{Z}_q}\)

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Abstract

Quasi-cyclic (QC) codes are a remarkable generalization of cyclic codes. Many QC codes have been shown to be best for their parameters. In this paper, some structural properties of QC codes over the prime power integer residue ring \({\mathbb{Z}_q}\) are considered. An l-QC code of length lm over \({\mathbb{Z}_q}\) is viewed both as in the conventional row circulant form and also as a \({\frac{\mathbb{Z}_q[x]}{\langle x^m-1 \rangle}}\) -submodule of \({\frac{GR(q,l)[x]}{\langle x^m-1 \rangle}}\) , where GR(q, l) is the Galois extension ring of degree l over \({\mathbb{Z}_q}\) . A necessary and sufficient condition for cyclic codes over Galois rings to be free is obtained and a BCH type bound for them is also given. A sufficient condition for 1-generator QC codes to be \({\mathbb{Z}_q}\) -free is given and a formula to evaluate their ranks is derived. Some distance bounds for 1-generator QC codes are also discussed. The duals of QC codes over \({\mathbb{Z}_q}\) are also briefly discussed.

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References

  1. Aydin N., Ray-Chaudhuri D.K.: Quasi-cyclic codes over \({\mathbb{Z}_4}\) and some new binary codes. IEEE Trans. Inf. Theory. 48, 2065–2069 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bini G., Flamini F.: Finite Commutative Rings and their Applications. Kluwer, Dordrecht (2002)

    MATH  Google Scholar 

  3. Blake I.F.: Codes over certain rings. Inf. Control. 20, 396–404 (1972)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  5. Conan J., Séguin G.: Structural properties and enumeration of quasi-cyclic codes. Appl. Algebra Eng. Commun. Comput. 4, 25–39 (1993)

    Article  MATH  Google Scholar 

  6. Daskalov R., Hristov P.: New binary one-generator quasi-cyclic codes. IEEE Trans. Inf. Theory. 49, 3001–3005 (2003)

    Article  MathSciNet  Google Scholar 

  7. Fossorier M.P.C.: Quasi-cyclic low-density parity-check codes from circulant permutation matrices. IEEE Trans. Inf. Theory. 50, 1788–1793 (2004)

    Article  MathSciNet  Google Scholar 

  8. Hammons A.R. Jr., Kumar P.V., Calderbank A.R., Sloane N.J.A., Solé P.: The \({\mathbb{Z}_4}\) linearity of Kerdock, Preparata, Goethals and related codes. IEEE Trans. Inf. Theory. 40, 301–319 (1994)

    Article  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  10. Kasami T.: A Gibert-Varshamov bound for quasi-cyclic codes of rate 1/2. IEEE Trans. Inf. Theory. 20, 679 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  11. Lally, K.: Quasi-cyclic codes of index l over \({\mathbb{F}_q}\) viewed as \({\mathbb{F}_q[x]}\) -submodules of \({\mathbb{F}_{q^l}[x]/\langle x^m -1 \rangle}\) . In: Fossorier, M., Hoeholdt, T., Poli, A. (Eds.) Proceedings of Applied Algebra, Algebraic Algorithm and Error Correcting Codes-15, LNCS- 2643, pp. 244–253. Springer-Verlag, Berlin, Heidelberg (2003)

  12. Lally K., Fitzpatrick P.: Algebraic structure of quasi-cyclic codes. Discrete Appl. Math. 111, 157–175 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  13. Ling S., Solé P.: On the algebraic structures of quasi-cyclic codes I: finite fields. IEEE Trans. Inf. Theory 47, 2751–2760 (2001)

    Article  MATH  Google Scholar 

  14. Ling S., Solé P.: On the algebraic structures of quasi-cyclic codes II: chain rings. Des. Codes Cryptogr. 30, 113–130 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  15. Ling S., Solé P.: On the algebraic structures of quasi-cyclic codes III: generator theory. IEEE Trans. Inf. Theory. 51, 2692–2700 (2005)

    Article  MATH  Google Scholar 

  16. Maheshanand, Wasan, S.K.: On quasi-cyclic codes over integer residue rings. In: Boztas, S., Lu, H.F. (Eds.) Proceedings of Applied Algebra, Algebraic Algorithms and Error Correcting Codes-17, LNCS-4851, pp. 330–336. Springer-Verlag, Berlin, Heidelberg (2007)

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

    MATH  Google Scholar 

  18. McGuire G.: An approach to Hensel’s lemma. Irish Math. Soc. Bull. 47, 15–21 (2001)

    MATH  MathSciNet  Google Scholar 

  19. Norton G.H., Sălăgean A.: On the structure of linear and cyclic codes over finite chain rings. Appl. Algebra Eng. Commun. Comput. 10, 489–506 (2000)

    Article  MATH  Google Scholar 

  20. Pless V.S., Qian Z.: Cyclic codes and quadratic residue codes over \({\mathbb{Z}_4}\) . IEEE Trans. Inf. Theory. 42, 1594–1600 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  21. Séguin G.E.: A class of 1-generator quasi-cyclic codes. IEEE Trans. Inf. Theory. 50, 1745–1753 (2004)

    Article  Google Scholar 

  22. Shankar P.: On BCH codes over arbitrary integer rings. IEEE Trans. Inf. Theory. 25, 480–483 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  23. Siap I., Aydin N., Ray-Chaudhuri D.K.: New ternary quasi-cyclic codes with better minimum distances. IEEE Trans. Inf. Theory. 46, 1554–1558 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  24. Speigel E.: Codes over \({\mathbb{Z}_m}\) . Inf. Control. 35, 48–51 (1977)

    Article  Google Scholar 

  25. Tanner, R.M.: Towards algebraic theory of Turbo codes. Proceedings of 2nd International Symposium on Turbo codes, Brest, France, 17–26 (2000)

  26. van Tilborg H.C.A.: On quasi-cyclic codes with rate 1/m. IEEE Trans. Inf. Theory. 24, 628–630 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  27. Wan Z.X.: Quaternary Codes. World Scientific, Singapore (1997)

    MATH  Google Scholar 

  28. Wan Z.X.: Cyclic codes over Galois rings. Algebra Colloq. 6, 291–304 (1999)

    MATH  MathSciNet  Google Scholar 

  29. Wan Z.X.: Lectures on Finite Fields and Galois Rings. World Scientific, Singapore (2003)

    MATH  Google Scholar 

  30. Wasan S.K.: On codes over \({\mathbb{Z}_m}\) . IEEE Trans. Inf. Theory. 28, 117–120 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  31. Woo S.S.: Free cyclic codes over finite local rings. Bull. Korean Math. Soc. 43, 723–735 (2006)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Centre for Development of Advanced Computing, B-30, Sector-62, Noida, 201307, India

    Maheshanand Bhaintwal

  2. Department of Mathematics, Jamia Millia Islamia, New Delhi, 110025, India

    Siri Krishan Wasan

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  1. Maheshanand Bhaintwal
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  2. Siri Krishan Wasan
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Correspondence to Maheshanand Bhaintwal.

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Bhaintwal, M., Wasan, S.K. On quasi-cyclic codes over \({\mathbb{Z}_q}\) . AAECC 20, 459–480 (2009). https://doi.org/10.1007/s00200-009-0110-8

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  • DOI: https://doi.org/10.1007/s00200-009-0110-8

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