New Constructions of Group-Invariant Butson Hadamard Matrices.

Let G be a finite group and let h be a positive integer. A BH(G, h) matrix is a G-invariant ∣G∣ × ∣G∣ matrix H whose entries are complex hth roots of unity such that H H* = ∣G∣I_∣G∣, where H* denotes the complex conjugate transpose of H, and I_∣G∣ denotes the identity matrix of order ∣G∣. In this paper, we give three new constructions of BH(G, h) matrices. The first construction is the first known family of BH(G, h) matrices in which G does not need to be abelian. The second and the third constructions are two families of BH(G, h) matrices in which G is a finite local ring. [ABSTRACT FROM AUTHOR]