Cyclic codes with few weights and Niho exponents (Q703686)

In this paper binary cyclic codes which have two nonzeros only are treated. Such a code is of length \(n=2^{m}-1\), \(m=2t\), and of dimension \(2m\). To study its weights is actually to compute the sums \(S_{k}(a)=\sum _{x\in F_{2^{m}}} (-1)^{\text{Tr}(x^{k}+ax)}\), \(a\in F_{2^{m}}\), where \(k\) and \(n\) are coprime and Tr is the trace function on \( F_{2^{m}}\), the finite field of order \(2^{m}\). These sums can be seen as the Fourier-transforms of the Boolean function Tr\((x^{k})\) and the function \(S_{k}(a)\) provides the crosscorrelation function of one \(m\)-sequence of length \(n\) with its decimation by \(k\). The main result is that when \(k\equiv 2^{j}\) (mod \(2^{t}-1\)), for some \(j\), then \(S_{k}(a)\) takes at least four values when \(a\) runs through \( F_{2^{m}}\). This produces an upper bound on the nonlinearity of these Boolean functions, but to find the exact nonlinearity remains an open problem. Finally, three conjectures are proposed.