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Bounds on quasi-cyclic codes over finite chain rings

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Abstract

In this short correspondence, we mainly consider quasi-cyclic (QC) codes over finite chain rings. We study module structures and trace representations of QC codes, which lead to some lower bounds on the minimum Hamming distance of QC codes.

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References

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

    Article  MathSciNet  MATH  Google Scholar 

  2. Bhaintwal, M., Wasan, S.: On quasi-cyclic codes over \(\mathbb{Z}_q\). Appl. Algebra Eng. Commun. Comput. 20, 459–480 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cao, Y., Gao, J.: Constructing quasi-cyclic codes from linear algebra theory. Des. Codes Crypt. 67, 59–75 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Güneri, C.: Artin–Schreier curves and weights of two-dimensional cyclic codes. Finite Fields Appl. 10, 481–505 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  5. Güneri, C., Özbudak, F.: A bound on the minimum distance of quasi-cyclic codes. SIAM J. Discret. Math. 26, 1781–1796 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Lally, K.: Quasicyclic codes of index \(\ell \) over \(\mathbb{F}_q\) viewed as \(\mathbb{F}_q[x]\)-submodules of \(\mathbb{F}_{q^\ell }[x]/(x^m-1)\), in applied algebra, algebraic algorithms and error-correcting codes. Lect. Notes Comput. Sci. 2643, 244–253 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ling, S., Solé, P.: On the algebra structure of quasi-cyclic codes II: chain rings. Des. Codes Crypt. 30, 113–130 (2003)

    Article  MATH  Google Scholar 

  8. MacDonald, B.: Finite Rings with Identity. Dekker, New York (1974)

    Google Scholar 

  9. Norton, G., Sâlâgean, A.: On the stucture of linear and cyclic codes over a finite chain ring. Appl. Algebra Eng. Commun. Comput. 6, 489–506 (2000)

    Article  MATH  Google Scholar 

  10. Siap, I., Abualrub, T., Yildiz, B.: One generator quasi-cyclic codes over \(\mathbb{F}_2+u\mathbb{F}_2\). J. Frank. Inst. 349, 284–292 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. Wan, Z.-X.: Cyclic codes over Galois rings. Algebra Colloq. 6(3), 291–304 (1999)

    MathSciNet  MATH  Google Scholar 

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Acknowledgments

This research is supported by the National Key Basic Research Program of China (Grant No. 2013CB834204), and the National Natural Science Foundation of China (Grant Nos. 61171082 and 61301137).

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Authors and Affiliations

  1. Chern Institute of Mathematics and LPMC, Nankai University, Tianjin, People’s Republic of China

    Jian Gao, Linzhi Shen & Fang-Wei Fu

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  1. Jian Gao
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  2. Linzhi Shen
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  3. Fang-Wei Fu
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Correspondence to Jian Gao.

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Gao, J., Shen, L. & Fu, FW. Bounds on quasi-cyclic codes over finite chain rings. J. Appl. Math. Comput. 50, 577–587 (2016). https://doi.org/10.1007/s12190-015-0885-7

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  • DOI: https://doi.org/10.1007/s12190-015-0885-7

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