Experimental constructions of binary matrices with good peak-sidelobe distances

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Skirlo et al., in “Binary matrices of optimal autocorrelations as alignment marks” [J. Vac. Sci. Technol., B 33, 1 (2015)], 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 × 7, and optimal diagonally symmetric matrices of dimensions 8 × 8 and 9 × 9. The authors make an initial investigation into and propose a strategy for (deterministically) constructing binary matrices with good peak-sidelobe distances. The authors 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, the authors formulate a tight upper bound on the peak-sidelobe distance of a certain class of circulant matrices. Interestingly, binary matrices corresponding to certain difference sets and almost difference sets have peak-sidelobe distances meeting this upper bound.