Optimal quinary negacyclic codes with minimum distance four

Based on solutions of certain equations over finite yields, a necessary and sufficient condition for the quinary negacyclic codes with parameters \begin{document}$ [\frac{5^m-1}{2},\frac{5^m-1}{2}-2m,4] $\end{document} to have generator polynomial \begin{document}$ m_{\alpha^3}(x)m_{\alpha^e}(x) $\end{document} is provided. Several classes of new optimal quinary negacyclic codes with the same parameters are constructed by analyzing irreducible factors of certain polynomials over finite fields. Moreover, several classes of new optimal quinary negacyclic codes with these parameters and generator polynomial \begin{document}$ m_{\alpha}(x)m_{\alpha^e}(x) $\end{document} are also presented.