Another expression of the MacWilliams identities and its applications

Let \begin{document}$ \mathcal C $\end{document} be a maximum distance separable (MDS) linear code over finite field id="M2">\begin{document}$ \Bbb F_q $\end{document}. In this paper, we present new formula of its weight distribution, which can seen as another expression the MacWilliams identities. Moreover, using formula, obtain id="M3">\begin{document}$ b $\end{document}-symbol distribution id="M4">\begin{document}$ in cases: id="M5">\begin{document}$ = 2 and id="M6">\begin{document}$ 3 $\end{document}, respectively.