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A characterization of all semiprimitive irreducible cyclic codes in terms of their lengths

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Abstract

Just by looking at the lengths of irreducible cyclic codes, we present a simple numerical characterization by which we can easily identify those codes that are either one-weight or semiprimitive two-weight irreducible cyclic codes over any finite field. We then particularize our characterization to the class of irreducible cyclic codes of dimension two, and with this, we show that regardless of the finite field any code in this class is always either a one-weight or a semiprimitive two-weight irreducible cyclic code. We also explore the weight distribution of another kind of irreducible cyclic codes and present an infinite family comprising two subfamilies of irreducible cyclic codes that were recently studied.

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References

  1. Aubry, Y., Langevin, P.: On the weights of binary irreducible cyclic codes. In: Proceedings of Workshop on Coding and Cryptography, Bergen, Norway. Lecture Notes in Computer Science, vol. 3969, pp. 161–169 (2005)

  2. Aubry, Y., Langevin, P.: On the semiprimitivity of cyclic codes. Algebraic geometry and its applications. Ser. Number Theory Appl. 5, 284–293 (2008)

    MATH  Google Scholar 

  3. Baumert, L.D., McEliece, R.J.: Weights of irreducible cyclic codes. Inf. Contr. 20(2), 158–175 (1972)

    Article  MathSciNet  Google Scholar 

  4. Brochero Martínez, F.E., Giraldo Vergara, C.R.: Weight enumerator of some irreducible cyclic codes. Des. Codes Cryptogr. 78, 703–712 (2016)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  6. Ding, C.: The weight distribution of some irreducible cyclic codes. IEEE Trans. Inf. Theory 55(3), 955–960 (2009)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  8. Helleseth, T., Kløve, T., Mykkeltveit, J.: The weight distribution of irreducible cyclic codes with block lengths \(n_1((q^l-1)N)\). Discrete Math. 18(2), 179–211 (1977)

    Article  MathSciNet  Google Scholar 

  9. Kløve, T.: The weight distribution for a class of irreducible cyclic codes. Discrete Math. 20, 87–90 (1977)

    Article  MathSciNet  Google Scholar 

  10. Lidl, R., Niederreiter, H.: Finite Fields. Cambridge University Press, Cambridge (1983)

    MATH  Google Scholar 

  11. MacWilliams, F.J., Seery, J.: The weight distribution of some minimal cyclic codes. IEEE Trans. Inf. Theory IT–27, 796–801 (1981)

    Article  MathSciNet  Google Scholar 

  12. MacWilliams, F.J., Sloane, N.J.A.: The Theory of Error-Correcting Codes. North-Holland, Amsterdam (1977)

    MATH  Google Scholar 

  13. Rao, A., Pinnawala, N.: A family of two-weight irreducible cyclic codes. IEEE Trans. Inf. Theory 56, 2568–2570 (2010)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  15. Sharma, A., Bakshi, G.K.: The weight distribution of some irreducible cyclic codes. Finite Fields Appl. 18(1), 144–159 (2012)

    Article  MathSciNet  Google Scholar 

  16. Shi, M., Zhang, Z., Solé, P.: Two-weight codes and second order recurrences. https://arxiv.org/pdf/1709.04585.pdf (2017)

  17. Vega, G.: A critical review and some remarks about one- and two-weight irreducible cyclic codes. Finite Fields Appl. 51(33), 1–13 (2015)

    Article  MathSciNet  Google Scholar 

  18. Vega, G.: An improved method for determining the weight distribution of a family of q-ary cyclic codes. Appl. Algebra Eng. Commun. Comput. 28, 527–533 (2017)

    Article  MathSciNet  Google Scholar 

  19. Vega, G.: A correction on the determination of the weight enumerator polynomial of some irreducible cyclic codes. Des. Codes Cryptogr. 86(4), 835–840 (2018)

    Article  MathSciNet  Google Scholar 

  20. Vega, G.: The weight distribution for any irreducible cyclic code of length \(p^m\). Appl. Algebra Eng. Commun. Comput. 29, 363–370 (2018)

    Article  Google Scholar 

  21. Wolfmann, J.: Are 2-weight projective cyclic codes irreducible? IEEE Trans. Inf. Theory 51(2), 733–737 (2005)

    Article  MathSciNet  Google Scholar 

  22. Wolfmann, J.: Projective two-weight irreducible cyclic and constacyclic codes. Finite Fields Appl. 14(2), 351–360 (2008)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The author want to express his gratitude to the anonymous referees for their valuable suggestions.

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  1. Dirección General de Cómputo y de Tecnologías de Información y Comunicación, Universidad Nacional Autónoma de México, 04510, Ciudad de México, Mexico

    Gerardo Vega

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  1. Gerardo Vega
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Correspondence to Gerardo Vega.

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Vega, G. A characterization of all semiprimitive irreducible cyclic codes in terms of their lengths. AAECC 30, 441–452 (2019). https://doi.org/10.1007/s00200-019-00385-z

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  • DOI: https://doi.org/10.1007/s00200-019-00385-z

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