On the cross-correlation properties of large-size families of Costas arrays
Runfeng Liu, Qi Wang
TL;DR
The paper addresses the cross-correlation problem for large families of Costas arrays, extending beyond classical auto-correlation properties. By transforming cross-correlation calculations into polynomial and fixed-point counting problems, it derives explicit upper bounds for several large families, including the merged Welch family $\mathcal{W}^{el}_p$, power-permutation families $\mathcal{P}_p$, and their unions $\mathcal{PW}_p$ and $\mathcal{PW}^l_p$, with distinctions based on whether $p$ is a safe prime. Notably, it provides the first nontrivial bound for $\mathcal{PW}_p$ when $u$ varies and $v=0$, and a bound for $\mathcal{P}_p$ when $u=0$ and $v\neq 0$, leveraging Weil bounds and trinomial root estimates. The results enhance understanding of cross-interference in multi-user radar/communication contexts and open avenues for tighter bounds and new large-size Costas families. Overall, the work integrates algebraic constructions with analytic bounds to advance the design of Costas arrays with favorable cross-correlation properties.
Abstract
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 crosscorrelation are partially bounded for some cases of horizontal shifts $u$ and vertical shifts $v$. Given a prime $p \geq 5$, a large-size family of Costas arrays over $\{1, \ldots, p-1\}$ is investigated, including both the exponential 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 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 $v=0$ that it equals $(p-1) / t$ where $t$ is the smallest prime divisor of $(p-1) / 2$ if p is not a safe prime and is at most $(p-1)^{\frac{1}{2}}+(p-1)^{\frac{1}{4}}+\frac{1}{2}$ otherwise.