Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems, and communication systems due to their efficient encoding and decoding algorithms. In this paper, we try to construct the binary cyclic codes from sparse (permutation) polynomials over \(\mathbb {F}_{2^n}\) through the Ding’s method (Ding in SIAM J. Discret. Math. 27:1977–1994, 2013). As a consequence, we present several infinite families of binary cyclic codes of length \(2^n-1\) with dimensions larger than \(\frac{2^n-1}{2}\) , and characterize their minimum distance by using the sphere packing, Singleton, BCH and Hartmann-Tzeng bounds. Some of them are optimal or almost optimal in the sense that they attain the known bounds on cyclic codes. Specifically, we obtain four infinite families of optimal codes: two with parameters \([2^n-1, 2^n -3n-2, 8]\) and two with parameters \([2^n-1, 2^n -n-2, 4]\) .
