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

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

External links

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