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,

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

Existence

If p is prime then $H(p,N)$ can exist only for $N = mp$ with integer m and it is conjectured they exist for all such cases with $p \ge 3$. 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 $[F_N]_{jk}:= \exp[(2\pi i(j - 1)(k - 1) / N] {\quad \rm for \quad} j,k=1,2,\dots,N$

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, Canad. 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