Experimental Constructions of Binary Matrices with Good Peak-Sidelobe Distances

TL;DR

This paper explores deterministic methods to construct binary matrices with high peak-sidelobe autocorrelation distances, compares them to optimal matrices, and establishes bounds for circulant matrices, advancing design strategies for autocorrelation properties.

Contribution

It introduces new deterministic constructions for binary matrices with good autocorrelation properties and derives upper bounds for circulant matrices, extending prior exhaustive search results.

Findings

Constructed matrices are near optimal for small dimensions.

Established a tight upper bound for certain circulant matrices.

Matrices from difference sets meet the upper bound.

Abstract

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.