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

Let p be an odd prime, and m , k be positive integers with m ≥ 3 k . Let C 1 and C 2 be cyclic codes over F p with parity-check polynomials h 2 ( x ) h 3 ( x ) and h 1 ( x ) h 2 ( x ) h 3 ( x ) , respectively, where h 1 ( x ) , h 2 ( x ) and h 3 ( x ) are the minimal polynomials of γ − 1 , γ − ( p k + 1 ) and γ − ( p 3 k + 1 ) over F p , respectively, for a primitive element γ of F p m . Recently, Zeng et al. (2010) obtained the weight distribution of C 2 for m gcd ( m , k ) being odd. In this paper, we determine the weight distribution of C 1 , and the weight distribution of C 2 for the case that m gcd ( m , k ) is even. [ABSTRACT FROM AUTHOR]