Two classes of cyclic extended double-error-correcting Goppa codes

Abstract

<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M1">\begin{document}$ \Bbb F_{2^m} $\end{document}</tex-math></inline-formula> be a finite extension of the field <inline-formula><tex-math id="M2">\begin{document}$ \Bbb F_2 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ g(x) = x^2+\alpha x+1 $\end{document}</tex-math></inline-formula> a quadratic polynomial over <inline-formula><tex-math id="M4">\begin{document}$ \Bbb F_{2^m} $\end{document}</tex-math></inline-formula>. In this paper, two classes of cyclic extended double-error-correcting Goppa codes are proposed. We obtain the following two classes of Goppa codes: (1) cyclic extended Goppa code with the irreducible polynomial <inline-formula><tex-math id="M5">\begin{document}$ g(x) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ L = \Bbb F_{2^m}\cup \{\infty\} $\end{document}</tex-math></inline-formula>; (2) cyclic extended Goppa code with the reducible polynomial <inline-formula><tex-math id="M7">\begin{document}$ g(x) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M8">\begin{document}$ |L'| = 2^m-1 $\end{document}</tex-math></inline-formula>. In addition, the parameters of above cyclic extended Goppa codes are given.</p>