Constructions of $q$-ary Golay Complementary Pairs Over Flexible Non-Power-of-Two Lengths

Abstract

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 ${\text{A}}_i,{\text{B}}_i\in \{{\xi }^a:0\le a\le q-1\}$ with $\xi =e^{\frac{2\pi \sqrt{-1}}{q}}$.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 $2^{m}M$ for all integers $h,m\ge 1$. 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.