493 research outputs found

    On a conjecture of Helleseth

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    We are concern about a conjecture proposed in the middle of the seventies by Hellesseth in the framework of maximal sequences and theirs cross-correlations. The conjecture claims the existence of a zero outphase Fourier coefficient. We give some divisibility properties in this direction

    On a Conjecture of Helleseth

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    Abstract. We are concerned about a conjecture proposed in the middle of the seventies by Hellesseth in the framework of maximal sequences and theirs cross-correlations. The conjecture claims the existence of a zero outphase Fourier coefficient. We give some divisibility properties in this direction

    Divisibility of Weil Sums of Binomials

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    Consider the Weil sum WF,d(u)=∑x∈Fψ(xd+ux)W_{F,d}(u)=\sum_{x \in F} \psi(x^d+u x), where FF is a finite field of characteristic pp, ψ\psi is the canonical additive character of FF, dd is coprime to ∣F∗∣|F^*|, and u∈F∗u \in F^*. We say that WF,d(u)W_{F,d}(u) is three-valued when it assumes precisely three distinct values as uu runs through F∗F^*: this is the minimum number of distinct values in the nondegenerate case, and three-valued WF,dW_{F,d} are rare and desirable. When WF,dW_{F,d} is three-valued, we give a lower bound on the pp-adic valuation of the values. This enables us to prove the characteristic 33 case of a 1976 conjecture of Helleseth: when p=3p=3 and [F:F3][F:{\mathbb F}_3] is a power of 22, we show that WF,dW_{F,d} cannot be three-valued.Comment: 11 page

    Proofs of two conjectures on ternary weakly regular bent functions

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    We study ternary monomial functions of the form f(x)=\Tr_n(ax^d), where x\in \Ff_{3^n} and \Tr_n: \Ff_{3^n}\to \Ff_3 is the absolute trace function. Using a lemma of Hou \cite{hou}, Stickelberger's theorem on Gauss sums, and certain ternary weight inequalities, we show that certain ternary monomial functions arising from \cite{hk1} are weakly regular bent, settling a conjecture of Helleseth and Kholosha \cite{hk1}. We also prove that the Coulter-Matthews bent functions are weakly regular.Comment: 20 page
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