On the cross-correlation properties of large-size families of Costas arrays

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 $(p-1)/2$.