Connection of $ p $-ary $ t $-weight linear codes to Ramanujan Cayley graphs with $ t+1 $ eigenvalues

We characterize the connection between \begin{document}$ p $\end{document} -ary linear codes and Ramanujan Cayley graphs. We explicitly determine an equivalence between \begin{document}$ t $\end{document} -weight linear codes over the finite field \begin{document}$ \Bbb F_p $\end{document} and Ramanujan Cayley graphs with \begin{document}$ t+1 $\end{document} eigenvalues. In particular, we get an explicit criterion on the equivalence between two-weight linear codes and Ramanujan strongly regular graphs with explicit parameters. Using this characterization, we construct several families of Ramanujan Cayley graphs with two or three eigenvalues from known linear codes with two or three weights, respectively.