具有良好峰值旁瓣距离的二元矩阵的实验构造
Experimental Constructions of Binary Matrices with Good Peak-Sidelobe Distances
AI总结：
本研究针对具有良好峰值旁瓣距离的二元矩阵构造问题，提出确定性构造策略，构造的小维度矩阵接近最优，还推导了一类循环矩阵的峰值旁瓣距离紧上界，且部分差集对应矩阵满足该上界。
摘要：
Skirlo等人在《作为对准标记的最优自相关二元矩阵》[《真空科学与技术期刊B辑》33(2) (2015) 1-7]中，通过最大化非周期自相关中的峰值旁瓣距离定义了一类新的二元矩阵，并通过穷举计算机搜索找到了维度最高为7×7的最优方阵，以及维度为8×8和9×9的最优对角对称矩阵。我们对（确定性地）构造具有良好峰值旁瓣距离的二元矩阵展开了初步研究，并提出了一种构造策略。我们构造了几类此类矩阵，并将它们的距离与Skirlo等人找到的最优矩阵的距离进行了比较。我们的构造方法生成的矩阵在小维度下接近最优。此外，我们推导了某一类循环矩阵的峰值旁瓣距离的紧上界。有趣的是，对应于某些差集和几乎差集的二元矩阵，其峰值旁瓣距离满足该上界。
英文摘要：
Skirlo et al., in 'Binary matrices of optimal autocorrelations as alignment marks' [Journal of Vacuum Science and Technology Series B 33(2) (2015) 1-7], defined a new class of binary matrices by maximizing the peak-sidelobe distances in the aperiodic autocorrelations and, by exhaustive computer searches, found the optimal square matrices of dimension up to 7 x 7, and optimal diagonally symmetric matrices of dimensions 8 x 8 and 9 x 9. We make an initial investigation into and propose a strategy for (deterministically) constructing binary matrices with good peak-sidelobe distances. We construct several classes of these and compare their distances to those of the optimal matrices found by Skirlo et al. Our constructions produce matrices that are near optimal for small dimension. Furthermore, we formulate a tight upper bound on the peak-sidelobe distance of a cer- tain class of circulant matrices. Interestingly, binary matrices corresponding to certain difference sets and almost difference sets have peak-sidelobe distances meeting this upper bound.
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