Butson-type Hadamard matrix

In mathematics, a complex Hadamard matrixH of size N with all its columns (rows) mutually orthogonal, belongs to the Butson-typeH(q, N) if all its elements are powers of q-th root of unity,

${\displaystyle (H_{jk})^{q}=1\quad {\text{for}}\quad j,k=1,2,\dots ,N.}$

If p is prime and ${\displaystyle N>1}$, then ${\displaystyle H(p,N)}$ can exist only for ${\displaystyle N=mp}$ with integerm and it is conjectured they exist for all such cases with ${\displaystyle p\geq 3}$. For ${\displaystyle p=2}$, the corresponding conjecture is existence for all multiples of 4. In general, the problem of finding all sets ${\displaystyle \{q,N\}}$ such that the Butson-type matrices ${\displaystyle H(q,N)}$ exist, remains open.

• ${\displaystyle H(2,N)}$ contains realHadamard matrices of size N,
• ${\displaystyle H(4,N)}$ contains Hadamard matrices composed of ${\displaystyle \pm 1,\pm i}$ – such matrices were called by Turyn, complex Hadamard matrices.
• in the limit ${\displaystyle q\to \infty }$ one can approximate all complex Hadamard matrices.
• Fourier matrices ${\displaystyle [F_{N}]_{jk}:=\exp[(2\pi i(j-1)(k-1)/N]{\text{ for }}j,k=1,2,\dots ,N}$

belong to the Butson-type,

${\displaystyle F_{N}\in H(N,N),}$

while

${\displaystyle F_{N}\otimes F_{N}\in H(N,N^{2}),}$

${\displaystyle F_{N}\otimes F_{N}\otimes F_{N}\in H(N,N^{3}).}$

${\displaystyle D_{6}:={\begin{bmatrix}1&1&1&1&1&1\\1&-1&i&-i&-i&i\\1&i&-1&i&-i&-i\\1&-i&i&-1&i&-i\\1&-i&-i&i&-1&i\\1&i&-i&-i&i&-1\\\end{bmatrix}}\in \,H(4,6)}$,

${\displaystyle S_{6}:={\begin{bmatrix}1&1&1&1&1&1\\1&1&z&z&z^{2}&z^{2}\\1&z&1&z^{2}&z^{2}&z\\1&z&z^{2}&1&z&z^{2}\\1&z^{2}&z^{2}&z&1&z\\1&z^{2}&z&z^{2}&z&1\\\end{bmatrix}}\in \,H(3,6)}$

where ${\displaystyle z=\exp(2\pi i/3).}$

• A. T. Butson, Generalized Hadamard matrices, Proc. Am. Math. Soc. 13, 894-898 (1962).
• A. T. Butson, Relations among generalized Hadamard matrices, relative difference sets, and maximal length linear recurring sequences, Can. J. Math. 15, 42-48 (1963).
• R. J. Turyn, Complex Hadamard matrices, pp. 435–437 in Combinatorial Structures and their Applications, Gordon and Breach, London (1970).

• Complex Hadamard Matrices of Butson type - a catalogue, by Wojciech Bruzda, Wojciech Tadej and Karol Życzkowski, retrieved October 24, 2006