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

Year 2017, Volume: 4 Issue: 3, 235 - 246, 15.09.2017

Augustine İ. Musukwa Kondwani Magamba John A. Ryan

https://doi.org/10.13069/jacodesmath.327368

Cited By: 5

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.

Keywords

Goppa codes, Extended codes, Irreducible Goppa codes, Equivalent codes

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.