The weight distributions of two classes of binary cyclic codes

•A certain system of algebraic equations over finite fields is solved.•A connection between the “type” and rank of the quadratic forms in certain case is applied.•The weight distribution of a class of binary cyclic codes is determined by using the connection.•We determine another class of binary cyclic codes by using the Pless power moment identities

For two positive integers m and k  , let CeCe be a class of cyclic code of length 2m−12m−1 over F2F2 with three nonzeros γ−1γ−1, γ−(2k+1)γ−(2k+1) and γ−(2ek+1)γ−(2ek+1) for e=2e=2 or 3, where γ   is a primitive element of F2mF2m. When mgcd⁡(m,k) is odd, Kasami in 1971 determined the weight distributions of cyclic codes C2C2 and C3C3, which is the same as that of the dual of the triple error-correcting BCH code. This paper obtains the weight distributions of C2C2 and C3C3 for the case of mgcd⁡(m,k) being even.