Paper 2026/099

Arithmetic autocorrelation of binary half-$\ell$-sequences with connection integer $p^{r}q^{s}$

Abstract

Half-$\ell$-sequences, as a extension of $\ell$-sequences, have attracted research interest over the past decade. The arithmetic correlation of half-$\ell$-sequences is known for connection integers of the form $p^r$. In this paper, we extend this result by deriving the arithmetic correlation for half-$\ell$-sequences with connection integers of the form $p^r q^s $. The results indicate that when $p\equiv -1 \pmod{8}$ and $q\equiv \pm 3 \pmod{8}$, the arithmetic autocorrelation can be determined by the number of odd integers in the cyclic subgroup generated by $2$ modulo $p$.