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Enumeration of extended irreducible binary Goppa codes of degree $2^{m}$ and length $2^{n}+1$

Year 2017, , 235 - 246, 15.09.2017
https://doi.org/10.13069/jacodesmath.327368

Abstract

Let $n$ be an odd prime and $m>1$ be a positive integer. We produce an upper bound on the number of inequivalent extended irreducible binary Goppa codes of degree $2^{m}$ and length $2^{n}+1$. Some examples are given to illustrate our results.

References

  • [1] T. P. Berger, Goppa and related codes invariant under a prescribed permutation, IEEE Trans. Inform. Theory 46(7) (2000) 2628–2633.
  • [2] C. L. Chen, Equivalent irreducible Goppa codes, IEEE Trans. Inform. Theory 24(6) (1978) 766–769.
  • [3] H. Dinh, C. Moore, A. Russell, McEliece and Niederreiter cryptosystems that resist quantum Fourier sampling attacks, In: Rogaway P. (eds) Advances in Cryptology – CRYPTO 2011. CRYPTO 2011. Lecture Notes in Computer Science 6841 (2011) 761–779.
  • [4] I. M. Isaacs, Algebra: A Graduate Text, Brooks/Cole, Pacific Grove, 1994.
  • [5] R. Lidl, H. Niederreiter, Introduction to Finite Fields and Their Applications, Cambridge University Press, London, 1994.
  • [6] S. Ling, C. Xing, Coding Theory; A First Course, Cambridge University Press, United Kingdom, 2004.
  • [7] K. Magamba, J. A. Ryan, Counting irreducible polynomials of degree $r$ over $F_{q^n}$ and generating Goppa codes using the lattice of subfields of $F_{q^{nr}}$ , J. Discrete Math. 2014 (2014) 1–4.
  • [8] J. A. Ryan, Irreducible Goppa Codes, Ph.D. Dissertation, University College Cork, 2004.
  • [9] J. A. Ryan, A new connection between irreducible and extended irreducible Goppa codes, Proc. SAMSA (2012) 152–154.
  • [10] J. A. Ryan, Counting extended irreducible binary quartic Goppa codes of length $2^n+1$, IEEE Trans. Inform. Theory 61(3) (2015) 1174–1178.
Year 2017, , 235 - 246, 15.09.2017
https://doi.org/10.13069/jacodesmath.327368

Abstract

References

  • [1] T. P. Berger, Goppa and related codes invariant under a prescribed permutation, IEEE Trans. Inform. Theory 46(7) (2000) 2628–2633.
  • [2] C. L. Chen, Equivalent irreducible Goppa codes, IEEE Trans. Inform. Theory 24(6) (1978) 766–769.
  • [3] H. Dinh, C. Moore, A. Russell, McEliece and Niederreiter cryptosystems that resist quantum Fourier sampling attacks, In: Rogaway P. (eds) Advances in Cryptology – CRYPTO 2011. CRYPTO 2011. Lecture Notes in Computer Science 6841 (2011) 761–779.
  • [4] I. M. Isaacs, Algebra: A Graduate Text, Brooks/Cole, Pacific Grove, 1994.
  • [5] R. Lidl, H. Niederreiter, Introduction to Finite Fields and Their Applications, Cambridge University Press, London, 1994.
  • [6] S. Ling, C. Xing, Coding Theory; A First Course, Cambridge University Press, United Kingdom, 2004.
  • [7] K. Magamba, J. A. Ryan, Counting irreducible polynomials of degree $r$ over $F_{q^n}$ and generating Goppa codes using the lattice of subfields of $F_{q^{nr}}$ , J. Discrete Math. 2014 (2014) 1–4.
  • [8] J. A. Ryan, Irreducible Goppa Codes, Ph.D. Dissertation, University College Cork, 2004.
  • [9] J. A. Ryan, A new connection between irreducible and extended irreducible Goppa codes, Proc. SAMSA (2012) 152–154.
  • [10] J. A. Ryan, Counting extended irreducible binary quartic Goppa codes of length $2^n+1$, IEEE Trans. Inform. Theory 61(3) (2015) 1174–1178.
There are 10 citations in total.

Details

Subjects Engineering
Journal Section Articles
Authors

Augustine İ. Musukwa This is me 0000-0001-8792-6954

Kondwani Magamba This is me 0000-0003-4025-9802

John A. Ryan This is me

Publication Date September 15, 2017
Published in Issue Year 2017

Cite

APA Musukwa, A. İ., Magamba, K., & Ryan, J. A. (2017). Enumeration of extended irreducible binary Goppa codes of degree $2^{m}$ and length $2^{n}+1$. Journal of Algebra Combinatorics Discrete Structures and Applications, 4(3), 235-246. https://doi.org/10.13069/jacodesmath.327368
AMA Musukwa Aİ, Magamba K, Ryan JA. Enumeration of extended irreducible binary Goppa codes of degree $2^{m}$ and length $2^{n}+1$. Journal of Algebra Combinatorics Discrete Structures and Applications. September 2017;4(3):235-246. doi:10.13069/jacodesmath.327368
Chicago Musukwa, Augustine İ., Kondwani Magamba, and John A. Ryan. “Enumeration of Extended Irreducible Binary Goppa Codes of Degree $2^{m}$ and Length $2^{n}+1$”. Journal of Algebra Combinatorics Discrete Structures and Applications 4, no. 3 (September 2017): 235-46. https://doi.org/10.13069/jacodesmath.327368.
EndNote Musukwa Aİ, Magamba K, Ryan JA (September 1, 2017) Enumeration of extended irreducible binary Goppa codes of degree $2^{m}$ and length $2^{n}+1$. Journal of Algebra Combinatorics Discrete Structures and Applications 4 3 235–246.
IEEE A. İ. Musukwa, K. Magamba, and J. A. Ryan, “Enumeration of extended irreducible binary Goppa codes of degree $2^{m}$ and length $2^{n}+1$”, Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 4, no. 3, pp. 235–246, 2017, doi: 10.13069/jacodesmath.327368.
ISNAD Musukwa, Augustine İ. et al. “Enumeration of Extended Irreducible Binary Goppa Codes of Degree $2^{m}$ and Length $2^{n}+1$”. Journal of Algebra Combinatorics Discrete Structures and Applications 4/3 (September 2017), 235-246. https://doi.org/10.13069/jacodesmath.327368.
JAMA Musukwa Aİ, Magamba K, Ryan JA. Enumeration of extended irreducible binary Goppa codes of degree $2^{m}$ and length $2^{n}+1$. Journal of Algebra Combinatorics Discrete Structures and Applications. 2017;4:235–246.
MLA Musukwa, Augustine İ. et al. “Enumeration of Extended Irreducible Binary Goppa Codes of Degree $2^{m}$ and Length $2^{n}+1$”. Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 4, no. 3, 2017, pp. 235-46, doi:10.13069/jacodesmath.327368.
Vancouver Musukwa Aİ, Magamba K, Ryan JA. Enumeration of extended irreducible binary Goppa codes of degree $2^{m}$ and length $2^{n}+1$. Journal of Algebra Combinatorics Discrete Structures and Applications. 2017;4(3):235-46.