Butson-type Hadamard matrix

From Wikipedia, the free encyclopedia

This article may require cleanup to meet Wikipedia's quality standards.
Please discuss this issue on the talk page or replace this tag with a more specific message. This article has been tagged since October 2006.

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,

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

1 Existence
2 Examples
3 References
4 External link

[edit] Existence

If p is prime then $H (p, N)$ can exist only for $N = m p$ 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.

[edit] 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

$F_N \otimes F_N \in H(N,N^2)$ ,

$F_N \otimes F_N\otimes F_N \in H(N,N^3)$ .

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

$S_{6} := \begin{bmatrix} 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & z & z & z^2 & z^2 \\ 1 & z & 1 & z^2&z^2 & z \\ 1 & z & z^2& 1& z & z^2 \\ 1 & z^2& z^2& z& 1 & z \\ 1 & z^2& z & z^2& z & 1 \\ \end{bmatrix} \in H(3,6)$ , where $z = exp(2π i / 3)$ .

[edit] 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).