Legendre $G$-array pairs and the theoretical unification of several $G$-array families

Abstract

We investigate how Legendre $G$-array pairs are related to several different perfect binary $G$-array families. In particular we study the relations between Legendre $G$-array pairs, Sidelnikov-Lempel-Cohn-Eastman ${\mathbb{Z}}_{q-1}$-arrays, Yamada-Pott $G$-array pairs, Ding-Helleseth-Martinsen ${\mathbb{Z}}_2\times {\mathbb{Z}}_p^m$-arrays, Yamada ${\mathbb{Z}}_{(q-1)/2}$-arrays, Szekeres ${\mathbb{Z}}_p^m$-array pairs, Paley ${\mathbb{Z}}_p^m$-array pairs, and Baumert ${\mathbb{Z}}_{p_1}^{m_1}\times {\mathbb{Z}}_{p_2}^{m_2}$-array pairs. Our work also solves one of the two open problems posed in Ding~[J. Combin. Des. 16 (2008), 164-171]. Moreover, we provide several computer search based existence and non-existence results regarding Legendre ${\mathbb{Z}}_n$-array pairs. Finally, by using cyclotomic cosets, we provide a previously unknown Legendre ${\mathbb{Z}}_{57}$-array pair.