Cyclic codes with few weights and Niho exponents

This paper studies the values of the sums S(k)(a) = Sigma(xis an element ofF2mY) (-1)(Tr(Xk+ax)), a is an element of F(2)m, where Tr is the trace function on F(2)m = 2t and gcd(2(m) - 1, k) = 1. We mainly prove that when k drop 2(j) (mod 2(t) - 1), for somej, 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. (C) 2004 Elsevier Inc. All rights reserved.