This work is motivated by the long-standing open problem of designing asymptotically order-optimal aperiodic polyphase sequence sets with respect to the celebrated Welch bound. Attempts were made by Mow over 30 years ago, but a comprehensive understanding to this problem is lacking. Our first key contribution is an explicit upper bound of generalized quadratic Gauss sums which is obtained by recursively applying Paris' asymptotic expansion and then bounding it by leveraging the fast convergence property of the Fibonacci zeta function. Building upon this major finding, our second key contribution includes four systematic constructions of order-optimal sequence sets with low aperiodic correlation and/or ambiguity properties via carefully selected Chu sequences and Alltop sequences. For the first time in the literature, we reveal that the full Alltop sequence set is asymptotically optimal for its low aperiodic correlation sidelobes. Besides, we introduce a novel subset of Alltop sequences possessing both order-optimal aperiodic correlation and ambiguity properties for the entire time-shift window.

翻译：本工作的动机源于设计关于著名Welch界渐近阶数最优的非周期多相序列集这一长期悬而未决的开放性问题。Mow在三十多年前曾进行过尝试，但对该问题仍缺乏全面理解。我们的第一个关键贡献是获得了广义二次高斯和的显式上界，该结果通过递归应用Paris渐近展开式，并利用斐波那契zeta函数的快速收敛性进行界定而得到。基于这一重要发现，我们的第二个关键贡献包括通过精心选取Chu序列和Alltop序列，系统性地构建了四种具有低非周期相关性和/或模糊度特性的阶数最优序列集。我们首次在文献中揭示了完整Alltop序列集因其低非周期相关旁瓣而具有渐近最优性。此外，我们引入了一个新颖的Alltop序列子集，该子集在整个时移窗口内同时具备阶数最优的非周期相关性与模糊度特性。