Costas arrays have been an interesting combinatorial object for decades because of their optimal aperiodic auto-correlation properties. Meanwhile, it is interesting to find families of Costas arrays or extended arrays with small maximal cross-correlation values, since for applications in multi-user systems, the cross-interferences between different signals should also be small. The objective of this paper is to study several large-size families of Costas arrays or extended arrays, and their values of maximal cross-correlation are partially bounded for some cases of horizontal shifts $u$ and vertical shifts $v$. Given a prime $p \geq 5$, in particular, a large-size family of Costas arrays over $\{1, \ldots, p-1\}$ is investigated, including both the exponential Welch Costas arrays and logarithmic Welch Costas arrays. An upper bound on the maximal cross-correlation of this family for arbitrary $u$ and $v$ is given. We also show that the maximal cross-correlation of the family of power permutations over $\{1, \ldots, p-1\}$ for $u = 0$ and $v \neq 0$ is bounded by $\frac{1}{2}+\sqrt{p-1}$. Furthermore, we give the first nontrivial upper bound of $(p-1)/t$ on the maximal cross-correlation of the larger family including both exponential Welch Costas arrays and power permutations over $\{1, \ldots, p-1\}$ for arbitrary $u$ and $v=0$, where $t$ is the smallest prime divisor of $(p-1)/2$.

翻译：科斯塔斯阵列因其最优的非周期自相关特性，数十年来一直是备受关注的组合对象。同时，寻找具有较小最大互相关值的科斯塔斯阵列族或扩展阵列族也具有重要意义，因为在多用户系统应用中，不同信号间的相互干扰也应尽可能小。本文旨在研究几个大规模科斯塔斯阵列族或扩展阵列族，并针对某些水平偏移 $u$ 和垂直偏移 $v$ 的情况，部分地界定了它们的最大互相关值。给定一个素数 $p \geq 5$，本文特别研究了一个定义在 $\{1, \ldots, p-1\}$ 上的大规模科斯塔斯阵列族，该族同时包含指数型韦尔奇科斯塔斯阵列和对数型韦尔奇科斯塔斯阵列。我们给出了该族对于任意 $u$ 和 $v$ 的最大互相关的一个上界。我们还证明了，对于 $u = 0$ 且 $v \neq 0$ 的情况，定义在 $\{1, \ldots, p-1\}$ 上的幂置换族的最大互相关以 $\frac{1}{2}+\sqrt{p-1}$ 为界。此外，对于任意 $u$ 且 $v=0$ 的情况，我们给出了一个更大的阵列族（包含定义在 $\{1, \ldots, p-1\}$ 上的指数型韦尔奇科斯塔斯阵列和幂置换）的最大互相关的第一个非平凡上界 $(p-1)/t$，其中 $t$ 是 $(p-1)/2$ 的最小素因子。