AI 中文总结
解决Kumar等人的开放问题，在特殊情况\(q\equiv2\pmod4\)且\(d\)为奇数时构造广义弯曲函数，对于满足条件的奇数\(d\)给出显式函数，还表明其傅里叶系数非单位根，否定相关问题。
AI 中文摘要
我们通过构造从\((\mathbb{Z}/q\mathbb{Z})^d\)到\(\mathbb{Z}/q\mathbb{Z}\)的广义弯曲函数，解决了Kumar、Scholtz和Welch（1985）的一个开放问题，此为他们论文未给出构造且后续四十年仅通过不存在性结果处理的特殊情况\(q\equiv2\pmod4\)且\(d\)为奇数。具体而言，对于每个满足\(p = 2^d - 1\)是梅森素数的奇数整数\(d\geq3\)，我们构造了一个从\((\mathbb{Z}/2p\mathbb{Z})^d\)到\(\mathbb{Z}/2p\mathbb{Z}\)的显式广义弯曲函数，特别地，得到了一个\([3,14]\)型函数。我们还表明这些广义弯曲函数的傅里叶系数不是单位根，这对Armario、Egan、Kharaghani和Ó~Catháin关于特征表弯曲向量的一个近期问题给出了否定答案。
英文摘要
We resolve an open problem of Kumar, Scholtz, and Welch (1985) by constructing generalized bent functions from $(\mathbb{Z}/q\mathbb{Z})^m$ to $\mathbb{Z}/q\mathbb{Z}$ in the exceptional case $q\equiv2\pmod4$ with $m$ odd, the case their paper left without a construction and which four decades of subsequent work had addressed only through nonexistence results. Concretely, for all integers $m\geq d\geq 2$ with $m=dr+2k$, $r\geq 1$, $k\geq 0$, we construct an explicit generalized bent function from $(\mathbb{Z}/q\mathbb{Z})^m$ to $\mathbb{Z}/q\mathbb{Z}$, where $q=2(2^d-1)$. We further show that, when $d\ge3$, these generalized bent functions have Fourier coefficients that are not roots of unity --- all of them when $r$ is odd --- which gives a negative answer to a recent question of Armario, Egan, Kharaghani, and Ó~Catháin about bent vectors for character tables.
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