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Cyclic codes with few weights and Niho exponents. (English) Zbl 1072.94016
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.

##### MSC:
 94B15 Cyclic codes 06E30 Boolean functions
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##### References:
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