Quaternary Legendre Pairs

TL;DR

This paper introduces quaternary Legendre pairs, extending the concept beyond binary pairs, and demonstrates their applications in constructing Hadamard matrices and generating sequences for various lengths.

Contribution

It presents the first definition of quaternary Legendre pairs, explores their relation to Hadamard matrices, and provides explicit constructions for multiple sequence lengths.

Findings

Quaternary Legendre pairs can exist for even lengths.

They are useful in constructing Hadamard matrices.

Explicit sequences are constructed for various lengths.

Abstract

We introduce quaternary Legendre pairs of length $\ell$. In contrast to binary Legendre pairs they can exist for even $\ell$ as well. First we show that they are pertinent to the construction of quaternary Hadamard matrices of order $2\ell+2$ and thus of binary Hadamard matrices of order $4\ell+4$. Then for a prime $p>2$ we present a construction of a pair of sequences of length $p$ from which we can derive quaternary Legendre pairs of length $\ell=2p$ by decompression for $p=3,5,7,13,19,31,41$. Moreover, we give also constructions of Legendre pairs of length $\ell$ for all remaining even $\ell\le 24$.