Butson-type Hadamard matrix

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

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

Existence

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

Examples

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

belong to the Butson-type,

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

while

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

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

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

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

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

References

• 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).

External links

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