Quaternary Legendre Pairs

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