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.