Aperiodic Complementary Quadruples of Binary Sequences

Abstract

We have carried out a large number of computer searches for the base sequences \(BS(n + 1, n)\) as well as for three important subsets known as Turyn sequences, normal sequences, and near-normal sequences. In the Appendix, we give an extensive list of \(BS(n + 1, n)\) for \(n \leq 32\). The existence question for Turyn sequences in \(BS(n + 1, n)\) was resolved previously for all \(n \leq 41\), and we extend this bound to \(n \leq 51\). We also show that the sets \(BS(n + 1, n)\) do not contain any normal sequences if \(n = 27\) or \(n = 28\). To each set \(BS(n + 1, n)\), we associate a finite graph \(\Gamma_{n}\), and determine these graphs completely for \(n \leq 27\). We show that \(BS(m,n) = \emptyset \quad \text{if} \quad m \geq 2n, \; n > 1, \; \text{and} \; m + n \; \text{is odd}\),
and we also investigate the borderline case \(m = 2n – 1\).