New families of quaternionic Hadamard matrices

A quaternionic Hadamard matrix (QHM) of order n is an \(n\times n\) $n\times n$ matrix H with non-zero entries in the quaternions such that \(HH^*=nI_n\) $H{H}^{\ast }=n{I}_n$ , where \(I_n\) $I_n$ and \(H^*\) $H^{\ast }$ 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.