Quaternary Legendre pairs II
Ilias S. Kotsireas, Christoph Koutschan, Arne Winterhof
TL;DR
The paper advances the construction of quaternary Legendre pairs for even lengths by introducing a decomposition strategy that separates even and odd index subsequences, significantly reducing search complexity. It develops PSD-based pruning augmented by cyclotomic-field norms and Galois-theoretic criteria to filter candidates, and demonstrates the approach with explicit constructions for $\\ell=28,30,32,34$. The results push the smallest unresolved length beyond $\\ell=34$, provide a catalog of new qLPs, and connect to parallel work by Jedwab and Pender, highlighting the practical impact for quaternary Hadamard matrices and related coding/communication applications. The methods combine combinatorial search with algebraic number theory to yield rigorous necessary conditions and efficient enumeration of viable sequences. Overall, the paper delivers concrete new qLP instances, refines search techniques, and broadens the toolbox for constructing Hadamard-related sequences in the quaternary setting.
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.