Golay complementary pair (GCP), first introduced by Golay in 1951, has been extensively studied and widely applied in communication systems. A $q$-ary GCP $\{\mathbf{A},\mathbf{B}\}$ consists of two $q$-ary complex sequences $\mathbf{A}=(A_0,\cdots,A_{M-1})$ and $\mathbf{B}=({B}_0,\cdots,{B}_{M-1})$ of equal length $M$, where $\textit{A}_i,\textit{B}_i\in\{ξ^a:0\leq a\leq q-1\}$ with $ξ=e^{\frac{2π\sqrt{-1}}{q}}$.In this paper,we prove that the existence of a quaternary ($q=4$) GCP of length $M$ is equivalent to the explicit constructibility of ($4h$)-ary GCPs of length $2^mM$ for all integers $h,m\geq1$. All proposed sequences are constructed via extended Boolean functions (EBFs), and the direct construction yields GCPs with more flexible length ranges than all previous relevant results.

翻译：Golay互补对（GCP）由Golay于1951年首次提出，已在通信系统中得到广泛研究和应用。一个$q$元GCP $\{\mathbf{A},\mathbf{B}\}$由两个长度为$M$的$q$元复数序列$\mathbf{A}=(A_0,\cdots,A_{M-1})$和$\mathbf{B}=({B}_0,\cdots,{B}_{M-1})$组成，其中$\textit{A}_i,\textit{B}_i\in\{ξ^a:0\leq a\leq q-1\}$且$ξ=e^{\frac{2π\sqrt{-1}}{q}}$。本文证明，长度为$M$的四元（$q=4$）GCP的存在性等价于对所有整数$h,m\geq1$，长度为$2^mM$的（$4h$）元GCP的显式可构造性。所有提出的序列均通过扩展布尔函数（EBF）构造，且直接构造方法得到的GCP长度范围比此前所有相关结果更为灵活。