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

TL;DR

This paper introduces a new method for constructing $q$-ary Golay complementary pairs of flexible lengths using extended Boolean functions, expanding the range of lengths beyond previous methods.

Contribution

It establishes the equivalence between quaternary GCP existence and the constructibility of GCPs of certain lengths, using explicit constructions via extended Boolean functions.

Findings

Constructs GCPs with more flexible lengths than previous methods.

Proves the equivalence between quaternary GCP existence and constructibility of GCPs of lengths multiple of 4.

Uses extended Boolean functions for direct construction of GCPs.

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 $\textit{A}_i,\textit{B}_i\in\{\xi^a:0\leq a\leq q-1\}$ 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 $2^mM$ 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.