标题： 从向量双弯函数得到的混合字符和以及具有小字母表大小的渐近最优复数码本 显示英文标题

标题： Hybrid Character Sums From Vectorial Dual-Bent Functions and Asymptotically Optimal Complex Codebooks With Small Alphabet Sizes

摘要： 混合特征和是在编码理论和序列设计中具有良好应用的重要一类指数和。 设 $\gf_{p^m}$ 为具有 $p^m$ 个元素的有限域，其中 $p$ 是一个素数， $m$ 是一个正整数。 令 $V_n^{(p)}$ 为 $\gf_p$ 上的 $n$ 维向量空间，其中 $p$ 为素数。 在本文中，我们研究了如下形式的混合特征和 \begin{eqnarray*} \sum_{x \in V_n^{(p)}}\psi\left(F(x)\right)\chi_1\left(a x\right), \end{eqnarray*} 其中 $F$ 是从 $V_n^{(p)}$ 到 $\gf_{p^m}$ 的函数，且 $a \in V_n^{(p)}$, $\psi$ 是 $\gf_{p^m}$ 的非平凡乘法特征，而 $\chi_1$ 是 $V_n^{(p)}$ 的规范加法特征。 如果 $F(x)$ 是一个向量双弯函数且 $a \in V_n^{(p)}\setminus \{0\}$，在某些条件下，我们确定它们的复数模或显式值，这将一些已知结果作为特例进行推广。 可以得出结论，来自向量双弯函数的混合特征和具有非常小的复数模。 作为应用，从向量双弯函数构造了三类渐近最优复数码本，并基于混合特征和确定了它们的最大互相关幅度。 所构造的码本具有非常小的字母表大小，这增强了它们的实用性。 此外，这三个码本族都只有两值或三值的互相关幅度。 显示英文摘要

摘要： Hybrid character sums are an important class of exponential sums which have nice applications in coding theory and sequence design. Let $\gf_{p^m}$ be the finite field with $p^m$ elements for a prime $p$ and a positive integer $m$. Let $V_n^{(p)}$ be an $n$-dimensional vector space over $\gf_p$ for a prime $p$. In this paper, we study the hybrid character sums of the form \begin{eqnarray*} \sum_{x \in V_n^{(p)}}\psi\left(F(x)\right)\chi_1\left(a x\right), \end{eqnarray*} where $F$ is a function from $V_n^{(p)}$ to $\gf_{p^m}$ and $a \in V_n^{(p)}$, $\psi$ is a nontrivial multiplicative character of $\gf_{p^m}$ and $\chi_1$ is the canonical additive character of $V_n^{(p)}$. If $F(x)$ is a vectorial dual-bent function and $a \in V_n^{(p)}\setminus \{0\}$, we determine their complex modulus or explicit values under certain conditions, which generalizes some known results as special cases. It is concluded that the hybrid character sums from vectorial dual-bent functions have very small complex modulus. As applications, three families of asymptotically optimal complex codebooks are constructed from vectorial dual-bent functions and their maximal cross-correlation amplitude are determined based on the hybrid character sums. The constructed codebooks have very small alphabet sizes, which enhances their appeal for implementation. Besides, all of the three families of codebooks have only two-valued or three-valued cross-correlation amplitudes.