More classes of optimal quinary cyclic codes of form $${\mathcal {C}}_{(1,e,s)}$$

Cyclic codes are an important subclass of linear codes. In this paper, we investigate the construction of quinary cyclic codes with parameters $$[5^{m}-1, 5^{m}-2m-2, 4]$$ and eight new classes optimal quinary cyclic codes of form $${\mathcal {C}}_{(1,e,s)}$$ are presented by discussing the solutions of certain equations over $${\mathbb {F}}_{5^{m}}$$ .