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 $n\le 32$. The existence question for Turyn sequences in $BS(n+1,n)$ was resolved previously for all $n\le 41$, and we extend this bound to $n\le 51$. 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 ${\mathrm{\Gamma }}_n$, and determine these graphs completely for $n\le 27$. We show that $BS(m,n)=\mathrm{\varnothing }\text{if}m\ge 2n,n>1,\text{and}m+n\text{is odd}$,
and we also investigate the borderline case $m=2n–1$.