Enumeration of extended irreducible binary Goppa codes of degree $2^{m}$ and length $2^{n}+1$
Keywords:
Goppa codes, Extended codes, Irreducible Goppa codes, Equivalent codes
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
T. P. Berger, Goppa and related codes invariant under a prescribed permutation, IEEE Trans. Inform. Theory 46(7) (2000) 2628–2633.
C. L. Chen, Equivalent irreducible Goppa codes, IEEE Trans. Inform. Theory 24(6) (1978) 766–769.
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.
I. M. Isaacs, Algebra: A Graduate Text, Brooks/Cole, Pacific Grove, 1994.
R. Lidl, H. Niederreiter, Introduction to Finite Fields and Their Applications, Cambridge University Press, London, 1994.
S. Ling, C. Xing, Coding Theory; A First Course, Cambridge University Press, United Kingdom, 2004.
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.
J. A. Ryan, Irreducible Goppa Codes, Ph.D. Dissertation, University College Cork, 2004.
J. A. Ryan, A new connection between irreducible and extended irreducible Goppa codes, Proc. SAMSA (2012) 152–154.
J. A. Ryan, Counting extended irreducible binary quartic Goppa codes of length $2^n+1$, IEEE Trans. Inform. Theory 61(3) (2015) 1174–1178.
C. L. Chen, Equivalent irreducible Goppa codes, IEEE Trans. Inform. Theory 24(6) (1978) 766–769.
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.
I. M. Isaacs, Algebra: A Graduate Text, Brooks/Cole, Pacific Grove, 1994.
R. Lidl, H. Niederreiter, Introduction to Finite Fields and Their Applications, Cambridge University Press, London, 1994.
S. Ling, C. Xing, Coding Theory; A First Course, Cambridge University Press, United Kingdom, 2004.
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.
J. A. Ryan, Irreducible Goppa Codes, Ph.D. Dissertation, University College Cork, 2004.
J. A. Ryan, A new connection between irreducible and extended irreducible Goppa codes, Proc. SAMSA (2012) 152–154.
J. A. Ryan, Counting extended irreducible binary quartic Goppa codes of length $2^n+1$, IEEE Trans. Inform. Theory 61(3) (2015) 1174–1178.
