Butson-type Hadamard matrix explained

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 for j,k = 1 , 2 , ... ,N.

Existence

If p is prime and

N> 1 , then H(p,N) can existonly for N = mp with integer m andit 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 matricesH(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 inf ty one can approximate all complex Hadamard matrices.

• Fourier matrices

[F _{N]} _{jk} : = \exp[( 2 \pi i (j -1 )(k -1 )/N] for j,k = 1 , 2 , ... ,N

belong to the Butson-type,

F _{N} \in H(N,N),

while

F _{N} ⊗ F _{N} \in H(N,N ^{2),} F _{N} ⊗ F _{N ⊗} 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