Cyclic codes with few weights and Niho exponents

This paper studies the values of the sums S k ( a ) = ∑ x ∈ F 2 m ( - 1 ) Tr ( x k + ax ) , a ∈ F 2 m , where Tr is the trace function on F 2 m , m = 2 t and gcd ( 2 m - 1 , k ) = 1 . We mainly prove that when k ≡ 2 j ( mod 2 t - 1 ) , for some j, then S k ( a ) takes at least four values when a runs through F 2 m . This result, and other derived properties, can be viewed in the study of weights of some cyclic codes and of crosscorrelation function of m-sequences.