Quaternary Legendre pairs II

TL;DR

This paper advances the construction of quaternary Legendre pairs for even lengths, providing new explicit examples for lengths 28 to 34 and introducing a novel search method that reduces computational complexity.

Contribution

It presents new constructions for quaternary Legendre pairs at specific even lengths and introduces a separation technique for subsequence search that enhances efficiency.

Findings

Constructed quaternary Legendre pairs for lengths 28, 30, 32, and 34.

Developed a new search algorithm separating even and odd index subsequences.

Improved the PSD test using Galois theory for cyclotomic fields.

Abstract

Quaternary Legendre pairs are pertinent to the construction of quaternary Hadamard matrices and have many applications, for example in coding theory and communications. In contrast to binary Legendre pairs, quaternary ones can exist for even length $\ell$ as well. It is conjectured that there is a quaternary Legendre pair for any even $\ell$. The smallest open case until now had been $\ell=28$, and $\ell=38$ was the only length $\ell$ with $28\le \ell\le 60$ resolved before. Here we provide constructions for $\ell=28,30,32$, and $34$. In parallel and independently, Jedwab and Pender found a construction of quaternary Legendre pairs of length $\ell=(q-1)/2$ for any prime power $q\equiv 1\bmod 4$, which in particular covers $\ell=30$, $36$, and $40$, so that now $\ell=42$ is the smallest unresolved case. The main new idea of this paper is a way to separate the search for the subsequences along even and odd indices which substantially reduces the complexity of the search algorithm. In addition, we use Galois theory for cyclotomic fields to derive conditions which improve the PSD test.