Two classes of cyclic extended double-error-correcting Goppa codes
Yanyan Gao, Qin Yue, Xinmei Huang, Yun Yang · Advances in Mathematics of Communications · 2022
Let \begin{document}$ \Bbb F_{2^m} $\end{document} be a finite extension of the field \begin{document}$ \Bbb F_2 $\end{document} and \begin{document}$ g(x) = x^2+\alpha x+1 $\end{document} a quadratic polynomial over \begin{document}$ \Bbb F_{2^m} $\end{document} . In this paper, two classes of cyclic extended double-error-correcting Goppa codes are proposed. We obtain the following two classes of Goppa codes: (1) cyclic extended Goppa code with the irreducible polynomial \begin{document}$ g(x) $\end{document} and \begin{document}$ L = \Bbb F_{2^m}\cup \{\infty\} $\end{document} ; (2) cyclic extended Goppa code with the reducible polynomial \begin{document}$ g(x) $\end{document} and \begin{document}$ |L'| = 2^m-1 $\end{document} . In addition, the parameters of above cyclic extended Goppa codes are given.