Year 2017, Volume 4 , Issue 3, Pages 235 - 246 2017-09-15

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

Augustine İ. MUSUKWA [1] , Kondwani MAGAMBA [2] , John A. RYAN [3]


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.
Goppa codes, Extended codes, Irreducible Goppa codes, Equivalent codes
  • [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.
Subjects Engineering
Journal Section Articles
Authors

Orcid: 0000-0001-8792-6954
Author: Augustine İ. MUSUKWA

Orcid: 0000-0003-4025-9802
Author: Kondwani MAGAMBA

Author: John A. RYAN

Dates

Publication Date : September 15, 2017

Bibtex @research article { jacodesmath327368, journal = {Journal of Algebra Combinatorics Discrete Structures and Applications}, issn = {}, eissn = {2148-838X}, address = {}, publisher = {Yildiz Technical University}, year = {2017}, volume = {4}, pages = {235 - 246}, doi = {10.13069/jacodesmath.327368}, title = {Enumeration of extended irreducible binary Goppa codes of degree \$2\^\{m\}\$ and length \$2\^\{n\}+1\$}, key = {cite}, author = {Musukwa, Augustine İ. and Magamba, Kondwani and Ryan, John A.} }
APA Musukwa, A , Magamba, K , Ryan, J . (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 . DOI: 10.13069/jacodesmath.327368
MLA Musukwa, A , Magamba, K , Ryan, J . "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 (2017 ): 235-246 <https://dergipark.org.tr/en/pub/jacodesmath/issue/30328/327368>
Chicago Musukwa, A , Magamba, K , Ryan, J . "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 (2017 ): 235-246
RIS TY - JOUR T1 - Enumeration of extended irreducible binary Goppa codes of degree $2^{m}$ and length $2^{n}+1$ AU - Augustine İ. Musukwa , Kondwani Magamba , John A. Ryan Y1 - 2017 PY - 2017 N1 - doi: 10.13069/jacodesmath.327368 DO - 10.13069/jacodesmath.327368 T2 - Journal of Algebra Combinatorics Discrete Structures and Applications JF - Journal JO - JOR SP - 235 EP - 246 VL - 4 IS - 3 SN - -2148-838X M3 - doi: 10.13069/jacodesmath.327368 UR - https://doi.org/10.13069/jacodesmath.327368 Y2 - 2016 ER -
EndNote %0 Journal of Algebra Combinatorics Discrete Structures and Applications Enumeration of extended irreducible binary Goppa codes of degree $2^{m}$ and length $2^{n}+1$ %A Augustine İ. Musukwa , Kondwani Magamba , John A. Ryan %T Enumeration of extended irreducible binary Goppa codes of degree $2^{m}$ and length $2^{n}+1$ %D 2017 %J Journal of Algebra Combinatorics Discrete Structures and Applications %P -2148-838X %V 4 %N 3 %R doi: 10.13069/jacodesmath.327368 %U 10.13069/jacodesmath.327368
ISNAD Musukwa, Augustine İ. , Magamba, Kondwani , Ryan, John A. . "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
AMA Musukwa A , Magamba K , Ryan J . 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): 235-246.
Vancouver Musukwa A , Magamba K , Ryan J . 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): 235-246.

Authors of the Article
Augustine İ. MUSUKWA [1]
Kondwani MAGAMBA [2]
John A. RYAN [3]