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 $\mathbb{Z}^m_{p}$-array pairs, Paley $\mathbb{Z}^m_{p}$-array pairs, and Baumert $\mathbb{Z}^{m_1}_{p_1}\times \mathbb{Z}^{m_2}_{p_2}$-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.

中文翻译：

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Legendre G-array对和几个G-array家族的理论统一

我们研究 Leg​​endre $G$-array 对如何与几个不同的完美二进制 $G$-array 家族相关。我们特别研究了 Legendre $G$-array 对、Sidelnikov-Lempel-Cohn-Eastman $\mathbb{Z}_{q-1}$-arrays、Yamada-Pott $G$-array 对、Ding- Helleseth-Martinsen $\mathbb{Z}_{2}\times \mathbb{Z}_p^{m}$-arrays, Yamada $\mathbb{Z}_{(q-1)/2}$-arrays, Szekeres $\mathbb{Z}^m_{p}$-array 对、Paley $\mathbb{Z}^m_{p}$-array 对和 Baumert $\mathbb{Z}^{m_1}_{p_1} \times \mathbb{Z}^{m_2}_{p_2}$-array 对。我们的工作还解决了 Ding 提出的两个开放问题之一~[J. 结合。德。16 (2008), 164-171]。此外，我们提供了几种基于计算机搜索的关于 Legendre $\mathbb{Z}_n$-array 对的存在和不存在结果。最后，通过使用分圆陪集，