Constructions of $q$-ary Golay Complementary Pairs Over Flexible Non-Power-of-Two Lengths
For communication systems requiring non-power-of-two length GCPs, this provides a constructive method that extends the known length range.
This paper proves that the existence of a quaternary Golay complementary pair (GCP) of length M implies the explicit constructibility of (4h)-ary GCPs of length 2^m M for all integers h,m ≥ 1, yielding GCPs with more flexible length ranges than previous results.
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.