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A new class of distance-optimal binary cyclic codes and their duals

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Applicable Algebra in Engineering, Communication and Computing Aims and scope

Abstract

Let \(m=8k\) and \(\alpha\) be a primitive element of the finite field \({{\mathbb {G}}{\mathbb {F}}}(2^m)\), where \(k\ge 2\) is an integer. In this paper, a class of binary cyclic codes \({{\mathcal {C}}}_{(u,v)}\) of length \(2^m-1\) with two nonzeros \(\alpha ^{-u}\) and \(\alpha ^{-v}\) is studied, where \((u,v)=(1,(2^{m}-1)/17)\). It turns out that \({{\mathcal {C}}}_{(u,v)}\) has parameters \([2^m-1,2^m-m-9,4]\) and is distance-optimal with respect to the Sphere Packing bound. The weight distribution of the dual of \({{\mathcal {C}}}_{(u,v)}\) is also completely determined based on some results on Gaussian periods.

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References

  1. Carlet, C., Ding, C., Yuan, J.: Linear codes from highly nonlinear functions and their secret sharing schemes. IEEE Trans. Inf. Theory 51(6), 2089–2102 (2005)

    Article  MATH  Google Scholar 

  2. Delsarte, P.: On subfield subcodes of modified Reed-Solomon codes. IEEE Trans. Inf. Theory 21(5), 575–576 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ding, C., Yang, J.: Hamming weights in irreducible cyclic codes. Discrete Math. 313(4), 434–446 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ding, C., Ling, S.: A \(q\)-polynomial approach to cyclic codes. Finite Fields Appl. 20(3), 1–14 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  5. Ding, C.: Cyclic codes from some monomials and trinomials. SIAM J. Discrete Math. 27(4), 1977–1994 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ding, C., Helleseth, T.: Optimal ternary cyclic codes from monomials. IEEE Trans. Inf. Theory 59(9), 5898–5904 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ding, C., Wang, X.: A coding theory construction of new systematic authentication codes. Theor. Comput. Sci. 330(1), 81–99 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. Ding, C., Yang, Y., Tang, X.: Optimal sets of frequency hopping sequences from linear cyclic codes. IEEE Trans. Inf. Theory 56(7), 3605–3612 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dobbertin, H., Helleseth, T., Kumar, V., Martinsen, H.: Ternary \(m\)-sequences with three-valued cross-correlation function: new decimations of Welch and Niho type. IEEE Trans. Inf. Theory 47(4), 1473–1481 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  10. Feng, T.: On cyclic codes of length \(2^{2^r}-1\) with two zeros whose dual codes have three weights. Des. Codes Cryptogr. 62(3), 253–258 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. Huffman, W.C., Pless, V.: Fundamentals of Error-Correcting Codes. Cambridge University Press, Cambridge (2003)

    Book  MATH  Google Scholar 

  12. Kløve, T.: Codes for Error Detection. World Scientific Publishing Co., Inc., New Jersey (2007)

    Book  MATH  Google Scholar 

  13. Li, C., Yue, Q., Li, F.: Weight distributions of cyclic codes with respect to pairwise coprime order elements. Finite Fields Appl. 28, 94–114 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  14. Li, C., Li, N., Helleseth, T., Ding, C.: The weight distributions of several classes of cyclic codes from APN monomials. IEEE Trans. Inf. Theory 60(8), 4710–4721 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. Li, F., Yue, Q., Li, C.: The minimum Hamming distances of irreducible cyclic codes. Finite Fields Appl. 29, 225–242 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  16. Li, N., Li, C., Helleseth, T., Ding, C., Tang, X.: Optimal ternary cyclic codes with minimum distance four and five. Finite Fields Appl. 30, 100–120 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. Luo, R., Wei, L., Cheng, F., Du, X.: A class of binary cyclic codes with four weights. IEICE Trans. Fundam. Electron. Commun. Comput. Sci. 100–A, 965–968 (2017)

    Article  Google Scholar 

  18. Lint, J.H.: Introduction to Coding Theory. Springer-Verlag, Berlin (1999)

    Book  MATH  Google Scholar 

  19. Pless, V.: Power moment identities on weight distributions in error correcting codes. Inf. Control 6, 147–152 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  20. Schmidt, B., White, C.: All two-weight irreducible cyclic codes. Finite Fields Appl. 8, 1–17 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  21. Yang, J., Xiong, M., Ding, C., Luo, J.: Weight distribution of a class of cyclic codes with arbitrary number of zeros. IEEE Trans. Inf. Theory 59(9), 5985–5993 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  22. Zheng, D., Wang, X., Hu, L., Zeng, X.: The weight distributions of two classes of \(p\)-ary cyclic codes. Finite Fields Appl. 29, 202–224 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  23. Zeng, X., Shan, J., Hu, L.: A triple-error-correcting cyclic code from the Gold and Kasami–Welch APN power functions. Finite Fields Appl. 18(1), 70–92 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  24. Zhou, Z., Ding, C.: A class of three-weight cyclic codes. Finite Fields Appl. 25, 79–93 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  25. Zhou, Z., Ding, C.: Seven classes of three-weight cyclic codes. IEEE Trans. Commun. 61(10), 4120–4126 (2013)

    Article  Google Scholar 

  26. Zeng, X., Fan, C., Zeng, Q., Qi, Y.: Two classes of binary cyclic codes and their weight distributions. Appl. Algebra Eng. Commun. Comput. (2019). https://doi.org/10.1007/s00200-019-00400-3

    Article  MATH  Google Scholar 

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Acknowledgements

The authors are very grateful to the Editor and the anonymous reviewers for careful reading and invaluable suggestions that improved the quality of this paper. This work was supported by the Natural Science Foundation of Shandong Province under Grant ZR2018LA001.

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Authors and Affiliations

  1. School of Mathematics, Southwest Jiaotong University, Chengdu, 610031, China

    Kaiqiang Liu & Jianpeng Wang

  2. School of Mathematics and Big Data, Dezhou University, Dezhou, 253023, China

    Wenli Ren & Feng Wang

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  1. Kaiqiang Liu
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  2. Wenli Ren
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  3. Feng Wang
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Correspondence to Wenli Ren.

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Liu, K., Ren, W., Wang, F. et al. A new class of distance-optimal binary cyclic codes and their duals. AAECC 34, 99–109 (2023). https://doi.org/10.1007/s00200-020-00478-0

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  • DOI: https://doi.org/10.1007/s00200-020-00478-0

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