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On the locality of quasi-cyclic codes over finite fields

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Abstract

A code is said to have locality r if any coordinate value in a codeword of that code can be recovered by at most r other coordinates. In this paper, we have studied the locality of quasi-cyclic codes over finite fields. The generator matrix of a quasi-cyclic code can be represented in the form of circulant matrices. We have obtained a bound on the locality of the code in terms of the weights of the associated polynomials to these circulant matrices. We have further analyzed the bounds on the locality, particularly in the case of 1-generator quasi-cyclic codes. An algorithm to find the locality of a quasi-cyclic code is also presented. We have given a construction of 1-generator quasi-cyclic codes with locality at most r using the zeros of its generator polynomial. Some examples have been given to illustrate the results presented in the paper.

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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(7), 2065–2069 (2002).

    Article  Google Scholar 

  2. Barbier M., Chabot C., Quintin G.: On quasi-cyclic codes as a generalization of cyclic codes. Finite Fields Appl. 18(5), 904–919 (2012).

    Article  MathSciNet  Google Scholar 

  3. Bhaintwal M., Wasan S.K.: On quasi-cyclic codes over \(\mathbb{Z}_q\). Appl. Algebra Eng. Commun. Comput. 20(5–6), 459–480 (2009).

    Article  Google Scholar 

  4. Bosma W., Cannon J., Playoust C.: The Magma algebra system. I. The user language. J. Symbolic Comput. 24(3–4), 235–265 (1997).

    Article  MathSciNet  Google Scholar 

  5. Chen C., Peterson W.W., Weldon E. Jr.: Some results on quasi-cyclic codes. Inf. Control 15(5), 407–423 (1969).

    Article  MathSciNet  Google Scholar 

  6. Conan J., Seguin G.: Structural properties and enumeration of quasi cyclic codes. Appl. Algebra Eng. Commun. Comput. 4(1), 25–39 (1993).

    Article  MathSciNet  Google Scholar 

  7. Gao J., Shen L., Fu F.W.: Bounds on quasi-cyclic codes over finite chain rings. J. Appl. Math. Comput. 50(1–2), 577–587 (2016).

    Article  MathSciNet  Google Scholar 

  8. Gopalan P., Huang C., Simitci H., Yekhanin S.: On the locality of codeword symbols. IEEE Trans. Inf. Theory 58(11), 6925–6934 (2012).

    Article  MathSciNet  Google Scholar 

  9. Goparaju S., Calderbank R.: Binary cyclic codes that are locally repairable. In: 2014 IEEE International Symposium on Information Theory, pp. 676–680. IEEE (2014).

  10. Gulliver T.A.: Construction of Quasi-Cyclic Codes. University of Victoria, Victoria (1989).

    Google Scholar 

  11. Heijnen P.W.: Some Classes of Linear Codes: Observations About Their Structure, Construction, (Non-) Existence and Decoding. Technische Universiteit Eindhoven, Eindhoven (1999).

    MATH  Google Scholar 

  12. Huang P., Yaakobi E., Uchikawa H., Siegel P.H.: Cyclic linear binary locally repairable codes. In: 2015 IEEE Information Theory Workshop (ITW), pp. 1–5 (2015).

  13. Jin L., Kan H., Zhang Y.: Constructions of locally repairable codes with multiple recovering sets via rational function fields. IEEE Trans. Inf. Theory 66, 202–209 (2020).

    Article  MathSciNet  Google Scholar 

  14. Ling S., Solé P.: On the algebraic structure of quasi-cyclic codes. I. Finite fields. IEEE Trans. Inf. Theory 47(7), 2751–2760 (2001).

    Article  MathSciNet  Google Scholar 

  15. Ling S., Solé P.: On the algebraic structure of quasi-cyclic codes II: chain rings. Des. Codes Cryptogr. 30(1), 113–130 (2003).

    Article  MathSciNet  Google Scholar 

  16. Pamies-Juarez L., Hollmann H.D., Oggier F.: Locally repairable codes with multiple repair alternatives. In: 2013 IEEE International Symposium on Information Theory, pp. 892–896. IEEE (2013).

  17. Papailiopoulos D.S., Dimakis A.G.: Locally repairable codes. IEEE Trans. Inf. Theory 60(10), 5843–5855 (2014).

    Article  MathSciNet  Google Scholar 

  18. Prakash N., Kamath G.M., Lalitha V., Kumar P.V.: Optimal linear codes with a local-error-correction property. In: 2012 IEEE International Symposium on Information Theory Proceedings, pp. 2776–2780. IEEE (2012).

  19. Rawat A.S., Papailiopoulos D.S., Dimakis A.G., Vishwanath S.: Locality and availability in distributed storage. IEEE Trans. Inf. Theory 62(8), 4481–4493 (2016).

    Article  MathSciNet  Google Scholar 

  20. Séguin G.E.: A class of 1-generator quasi-cyclic codes. IEEE Trans. Inf. Theory 50(8), 1745–1753 (2004).

    Article  MathSciNet  Google Scholar 

  21. Siap I., Aydin N., Ray-Chaudhuri D.K.: New ternary quasi-cyclic codes with better minimum distances. IEEE Trans. Inf. Theory 46(4), 1554–1558 (2000). https://doi.org/10.1109/18.850694.

    Article  MathSciNet  MATH  Google Scholar 

  22. Tamo I., Barg A.: A family of optimal locally recoverable codes. IEEE Trans. Inf. Theory 60(8), 4661–4676 (2014).

    Article  MathSciNet  Google Scholar 

  23. Tamo I., Barg A., Goparaju S., Calderbank R.: Cyclic LRC codes and their subfield subcodes. In: 2015 IEEE International Symposium on Information Theory (ISIT), pp. 1262–1266. IEEE (2015).

  24. Tamo I., Barg A., Goparaju S., Calderbank R.: Cyclic LRC codes, binary LRC codes, and upper bounds on the distance of cyclic codes. Int. J. Inf. Coding Theory 3(4), 345–364 (2016).

    MathSciNet  MATH  Google Scholar 

  25. Tan P., Zhou Z., Yan H., Parampalli U.: Optimal cyclic locally repairable codes via cyclotomic polynomials. IEEE Commun. Lett. 23(2), 202–205 (2018).

    Article  Google Scholar 

  26. Zeraatpisheh M., Esmaeili M., Gulliver T.A.: Quasi-cyclic codes: algebraic properties and applications. Comput. Appl. Math. 39(2), 1–21 (2020).

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This research is partially supported by Science and Engineering Research Board (SERB), India, under Grant No. CRG/2020/003785. The first author would like to thank the University Grants Commission, India, for providing financial support. Also, the authors would like to thank the anonymous referees for their careful reading of the manuscript and valuable comments.

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  1. Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, India

    Charul Rajput & Maheshanand Bhaintwal

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  1. Charul Rajput
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  2. Maheshanand Bhaintwal
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Correspondence to Maheshanand Bhaintwal.

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Communicated by K.-U. Schmidt.

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Rajput, C., Bhaintwal, M. On the locality of quasi-cyclic codes over finite fields. Des. Codes Cryptogr. 90, 759–777 (2022). https://doi.org/10.1007/s10623-022-01009-3

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  • DOI: https://doi.org/10.1007/s10623-022-01009-3

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