  In this paper, by analyzing solutions of certain equations in the finite
field $\mathbb{F}_{5^m}$, three classes of new optimal quinary cyclic codes
with parameters $[5^m-1,5^m-2m-2,4]$ and two theorems are presented. With the
help of the two theorems, perfect nonlinear monomials, almost perfect nonlinear
monomials and a number of other monomials are used to construct more classes of
new optimal quinary cyclic codes with the same parameters.
