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

Existence

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

Examples

  • H ( 2 , N ) {\displaystyle H(2,N)} contains real Hadamard matrices of size N,
  • H ( 4 , N ) {\displaystyle H(4,N)} contains Hadamard matrices composed of ± 1 , ± i {\displaystyle \pm 1,\pm i} – such matrices were called by Turyn, complex Hadamard matrices.
  • in the limit q → ∞ {\displaystyle q\to \infty } one can approximate all complex Hadamard matrices.
  • Fourier matrices [ F N ] j k := exp ⁡ [ ( 2 π i ( j − 1 ) ( k − 1 ) / N ]  for  j , k = 1 , 2 , … , N {\displaystyle _{jk}:=\exp{\text{ for }}j,k=1,2,\dots ,N}
belong to the Butson-type,
F N ∈ H ( N , N ) , {\displaystyle F_{N}\in H(N,N),}
while
F N ⊗ F N ∈ H ( N , N 2 ) , {\displaystyle F_{N}\otimes F_{N}\in H(N,N^{2}),}
F N ⊗ F N ⊗ F N ∈ H ( N , N 3 ) . {\displaystyle F_{N}\otimes F_{N}\otimes F_{N}\in H(N,N^{3}).}
D 6 := [ 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 ] ∈ H ( 4 , 6 ) {\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)} ,
S 6 := [ 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 ] ∈ H ( 3 , 6 ) {\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)}
where z = exp ⁡ ( 2 π i / 3 ) . {\displaystyle 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).

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