Two constructions of quaternary Legendre pairs of even length
Jonathan Jedwab, Thomas Pender
TL;DR
The authors establish two general methods to construct quaternary Legendre pairs of even length, addressing prior gaps in the theory. The First Construction yields a quaternary Legendre pair of length $(q-1)/2$ for every odd prime power $q \\equiv 1 \\pmod{4}$, while the Second Construction gives a pair of length $2p$ for odd primes $p$ with $2p-1$ a prime power, using Gray maps and amicable binary sets. These results connect to the broader framework of quaternary Hadamard matrices and provide concrete, infinitely extensible constructions (subject to certain number-theoretic conditions). The paper also supplies explicit sequence constructions, a worked example at $p=13$, and a detailed discussion of remaining open cases up to length $100$.
Abstract
We give the first general constructions of even length quaternary Legendre pairs: there is a quaternary Legendre pair of length $(q-1)/2$ for every prime power $q$ congruent to $1$ modulo $4$, and there is a quaternary Legendre pair of length $2p$ for every odd prime $p$ for which $2p-1$ is a prime power.