Butson-type Hadamard matrix

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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.

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[edit] 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.

[edit] Examples

  • H(4,N) contains Hadamard matrices composed of \pm 1, \pm i - such matrices were called by Turyn, 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(2πi / 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).
  • R. J. Turyn, Complex Hadamard matrices, pp. 435-437 in Combinatorial Structures and their Applications, Gordon and Breach, London (1970).

[edit] External links

Complex Hadamard Matrices of Butson type - a catalogue, by Wojciech Bruzda, Wojciech Tadej and Karol Życzkowski, retrieved October 24, 2006