Weil sums of binomials, three-level cross-correlation, and a conjecture of Helleseth
Let \(q\) be a power of a prime \(p\) and let \(\psi_q: \mathbb F_q\to\mathbb C\) be given by \(\psi_q(x)=\exp (2\pi iTr(x)/p)\). For an integer \(d\) prime to \(q-1\), the Weil sums studied here are: NEWLINE\[NEWLINE W_{q,d}(a)=\sum_{x\in\mathbb F_q} \psi_q(x^d+ax). NEWLINE\]NEWLINE The author considers two long-standing conjectures of Helleseth:NEWLINENEWLINE(A) If \(q>2\) and \(d\equiv 1\pmod{p-1}\) then \(W_{q,d}(a)=0\) for some \(a\in\mathbb F_q^*\).NEWLINENEWLINE(B) If \(q=p^{2^n}\) and \(d\) is non-degenerate then \(W_{q,d}\) takes at least four values over \(\mathbb F_q^*\).NEWLINENEWLINEThe main theorem here is: If \(W_{q,d}\) is 3-valued over \(\mathbb F_q^*\) then \(d\equiv 1\pmod{p-1}\) and \(W_{q,d}(a)\in\mathbb Z\) for all \(a\in \mathbb F_q\). This implies that, for fixed \(q\) and \(d\), at least one of (A) or (B) must hold. It is further shown that conjecture (B) holds for \(p=2\). A helpful appendix gives equivalent formulations in information theory (Walsh spectrum), cryptography (cross-correlations) and finite geometry (hyperplane intersections).
scientific article; zbMATH DE number 6071600
scientific article