Three new classes of optimal quinary cyclic codes with minimum distance four

Due to their wide applications in consumer electronics, data storage systems and communication systems, cyclic codes have been an important subject of study for many years. Recently, several classes of optimal quinary cyclic codes of the forms C(0,1,e) and C(1,e,s) are presented in the literature, where s = 5(m)-1/(2) and 2 = e = 5(m) - 2. In this paper, by considering the solutions of certain equations over finite fields, we give three new classes of infinite families of optimal quinary cyclic codes of the form C(1,e,s) with parameters [5m - 1, 5(m) - 2(m) - 2, 4] . Specifically, we make progress towards an open problem proposed by Gaofei Wu et al. [17].