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,

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

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 $p\ge 3$. 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 $q\to \mathrm{\infty }$ 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

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

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

$D_6:=\left[\begin{array}{cccccc}1& 1& 1& 1& 1& 1\\ 1& -1& i& -i& -i& i\\ 1& i& -1& i& -i& -i\\ 1& -i& i& -1& i& -i\\ 1& -i& -i& i& -1& i\\ 1& i& -i& -i& i& -1\end{array}\right]\in H(4,6)$,

$S_6:=\left[\begin{array}{cccccc}1& 1& 1& 1& 1& 1\\ 1& 1& z& z& z^2& z^2\\ 1& z& 1& z^2& z^2& z\\ 1& z& z^2& 1& z& z^2\\ 1& z^2& z^2& z& 1& z\\ 1& z^2& z& z^2& z& 1\end{array}\right]\in H(3,6)$

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

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