Skip to main content
SpringerLink
Log in

Some q-ary Cyclic Codes from Explicit Monomials over \(\mathbb{F}_{q}m\)

  • Coding Theory
  • Published:
Problems of Information Transmission Aims and scope Submit manuscript

Abstract

Cyclic codes as a subclass of linear codes have practical applications in communication systems, consumer electronics, and data storage systems due to their efficient encoding and decoding algorithms. The objective of this paper is to construct some cyclic codes by the sequence approach. More precisely, we determine the dimension and the generator polynomials of three classes of q-ary cyclic codes defined by some sequences with explicit polynomials over \(\mathbb{F}_{q}m\). The minimum distance of such cyclic codes is also discussed. Some of these codes are optimal according to code tables. Moreover, the third class of cyclic codes provides some answers for Open Problem 3 proposed by Ding and Zhou in [1].

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ding, C. and Zhou, Z., Binary Cyclic Codes from Explicit Polynomials over GF(2m), Discrete Math., 2014, vol. 321, pp. 76–89.

    Article  MathSciNet  Google Scholar 

  2. Grassl, M., Bounds on the Minimum Distance of Linear Codes and Quantum Codes (electronic tables). Available at http://www.codetables.de.

  3. Chien, R.T., Cyclic Decoding Procedures for Bose-Chaudhuri-Hocquenghem Codes, IEEE Trans. Inform. Theory, 1964, vol. 10, no. 4, pp. 357–363.

    Article  Google Scholar 

  4. Forney, G.D., Jr., On Decoding BCH Codes, IEEE Trans. Inform. Theory, 1965, vol. 11, no. 4, pp. 549–557.

    Article  MathSciNet  Google Scholar 

  5. Prange, E., Some Cyclic Error-Correcting Codes with Simple Decoding Algorithms, Tech. Rep. of Air Force Cambridge Research Center, Bedford, MA, USA, April, 1958, no. TN-58-156.

    Google Scholar 

  6. Berlekamp, E.R. and Justesen, J., Some Long Cyclic Linear Binary Codes Are Not So Bad, IEEE Trans. Inform. Theory, 1974, vol. 20, no. 3, pp. 351–356.

    Article  MathSciNet  Google Scholar 

  7. Calderbank, A.R., Li, W.-C.W., and Poonen, B., A 2-adic Approach to the Analysis of Cyclic Codes, IEEE Trans. Inform. Theory, 1997, vol. 43, no. 3, pp. 977–986.

    Article  MathSciNet  Google Scholar 

  8. Kai, X. and Zhu, S., On Cyclic Self-Dual Codes, Appl. Algebra Engrg. Comm. Comput., 2008, vol. 19, no. 6, pp. 509–525.

    Article  MathSciNet  Google Scholar 

  9. Li, C., Li, N., Helleseth, T., and Ding, C., The Weight Distributions of Several Classes of Cyclic Codes from APN Monomials, IEEE Trans. Inform. Theory, 2014, vol. 60, no. 8, pp. 4710–4721.

    Article  MathSciNet  Google Scholar 

  10. Liu, L., Xie, X., Li, L., and Zhu, S., The Weight Distributions of Two Classes of Nonbinary Cyclic Codes with Few Weights, IEEE Commun. Lett., 2017, vol. 21, no. 11, pp. 2336–2339.

    Article  Google Scholar 

  11. Luo, J. and Feng, K., Cyclic Codes and Sequences from Generalized Coulter-Matthews Function, IEEE Trans. Inform. Theory, 2008, vol. 54, no. 12, pp. 5345–5353.

    Article  MathSciNet  Google Scholar 

  12. Martinez-Pérez, C. and Willems, W., Is the Class of Cyclic Codes Asymptotically Good?, IEEE Trans. Inform. Theory, 2006, vol. 52, no. 2, pp. 696–700.

    Article  MathSciNet  Google Scholar 

  13. Moreno, O. and Kumar, P.V., Minimum Distance Bounds for Cyclic Codes and Deligne's Theorem, IEEE Trans. Inform. Theory, 1993, vol. 39, no. 5, pp. 1524–1534.

    Article  MathSciNet  Google Scholar 

  14. Rao, A. and Pinnawala, N., A Family of Two-Weight Irreducible Cyclic Codes, IEEE Trans. Inform. Theory, 2010, vol. 56, no. 6, pp. 2568–2570.

    Article  MathSciNet  Google Scholar 

  15. Roth, R.M. and Seroussi, G., On Cyclic MDS Codes of Length q over GF(q), IEEE Trans. Inform. Theory, 1986, vol. 32, no. 2, pp. 284–285.

    Article  MathSciNet  Google Scholar 

  16. van Lint, J.H. and Wilson, R.M., On the Minimum Distance of Cyclic Codes, IEEE Trans. Inform. Theory, 1986, vol. 32, no. 1, pp. 23–40.

    Article  MathSciNet  Google Scholar 

  17. Zeng, X., Hu, L., Jiang, W., Yue, Q., and Cao, X., The Weight Distribution of a Class of p-ary Cyclic Codes, Finite Fields Appl., 2010, vol. 16, no. 1, pp. 56–73.

    Article  MathSciNet  Google Scholar 

  18. Ding, C., Cyclic Codes from Some Monomials and Trinomials, SIAM J. Discrete Math., 2013, vol. 27, no. 4, pp. 1977–1994.

    Article  MathSciNet  Google Scholar 

  19. Ding, C., A Sequence Construction of Cyclic Codes over Finite Fields, Cryptogr. Commun., 2018, vol. 10, no. 2, pp. 319–341.

    Article  MathSciNet  Google Scholar 

  20. Rajabi, Z. and Khashyarmanesh, K., Some Cyclic Codes from Some Monomials, Appl. Algebra Engrg. Comm. Comput., 2017, vol. 28, no. 6, pp. 469–495.

    Article  MathSciNet  Google Scholar 

  21. Tang, C., Qi, Y., and Xu, M., A Note on Cyclic Codes from APN Functions, Appl. Algebra Engrg. Comm. Comput., 2014, vol. 25, no. 1–2, pp. 21–37.

    Article  MathSciNet  Google Scholar 

  22. Ding, C. and Helleseth, T., Optimal Ternary Cyclic Codes from Monomials, IEEE Trans. Inform. Theory, 2013, vol. 59, no. 9, pp. 5898–5904.

    Article  MathSciNet  Google Scholar 

  23. Huffman, W.C. and Pless, V., Fundamentals of Error-Correcting Codes, Cambridge: Cambridge Univ. Press, 2003.

    Book  Google Scholar 

  24. Ding, C., Xiao, G., and Shan, W., The Stability Theory of Stream Ciphers, Lect. Notes Comp. Sci., vol. 561, Heidelberg: Springer, 1991.

    Google Scholar 

  25. El Rouayheb, S.Y., Georghiades, C.N., Soljanin, E., and Sprintson, A., Bounds on Codes Based on Graph Theory, in Proc. 2007 IEEE Int. Sympos. on Information Theory (ISIT'2007), Nice, France, June 24–29, 2007, pp. 1876–1879.

    Chapter  Google Scholar 

  26. Lucas, E., Théorie des fonctions numériques simplement périodiques, Amer. J. Math., 1878, vol. 1, no. 4, pp. 289–321.

    Article  MathSciNet  Google Scholar 

  27. Dobbertin, H., Almost Perfect Nonlinear Power Functions over GF(2n): The Niho Case, Inform. and Comput., 1999, vol. 151, no. 1–2, pp. 57–72.

    Article  MathSciNet  Google Scholar 

  28. Hollmann, H.D.L. and Xiang, Q., A Proof of the Welch and Niho Conjectures on Cross-Correlations of Binary m-Sequences, Finite Fields Appl., 2001, vol. 7, no. 2, pp. 253–286.

    Article  MathSciNet  Google Scholar 

Download references

Funding

This research is supported in part by the National Natural Science Foundation of China under Projects 61772168, 61572168, and 11871187.

Author information

Authors and Affiliations

  1. School of Mathematics, Hefei University of Technology, Hefei, Anhui, P.R. China

    L. Li, S. Zhu, L. Liu & X. Kai

Authors
  1. L. Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. L. Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. X. Kai
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to L. Li.

Additional information

Russian Text © The Author(s), 2019, published in Problemy Peredachi Informatsii, 2019, Vol. 55, No. 3, pp. 60–82.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, L., Zhu, S., Liu, L. et al. Some q-ary Cyclic Codes from Explicit Monomials over \(\mathbb{F}_{q}m\) . Probl Inf Transm 55, 254–274 (2019). https://doi.org/10.1134/S0032946019030050

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0032946019030050

Key words

Search

Navigation