New families of quaternionic Hadamard matrices

A quaternionic Hadamard matrix (QHM) of order n is an n× n n × n matrix H with non-zero entries in the quaternions such that HH^*=nIₙ H H ∗ = n I n , where Iₙ I n and H^* H ∗ denote the identity matrix and the conjugate-transpose of H , respectively. A QHM is dephased if all the entries in its first row and first column are 1, and it is non-commutative if its entries generate a non-commutative group. The aim of our work is to provide new constructions of infinitely many (non-commutative dephased) QHMs; such matrices are used by Farkas et al. (IEEE Trans Inform Theory 69(6):3814–3824, 2023) to produce mutually unbiased measurements.