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,

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

Existence

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

Examples

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

belong to the Butson-type,

[math]\displaystyle{ F_N \in H(N,N), }[/math]

while

[math]\displaystyle{ F_N \otimes F_N \in H(N,N^2), }[/math]

[math]\displaystyle{ F_N \otimes F_N\otimes F_N \in H(N,N^3). }[/math]

[math]\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) }[/math],

[math]\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) }[/math]

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

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