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(qN) if all its elements are powers of q-th root of unity,

Existence

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If p is prime and  , then   can exist only for   with integer m and it is conjectured they exist for all such cases with  . For  , the corresponding conjecture is existence for all multiples of 4. In general, the problem of finding all sets   such that the Butson-type matrices   exist, remains open.

Examples

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  •   contains real Hadamard matrices of size N,
  •   contains Hadamard matrices composed of   – such matrices were called by Turyn, complex Hadamard matrices.
  • in the limit   one can approximate all complex Hadamard matrices.
  • Fourier matrices  
belong to the Butson-type,
 
while
 
 
 ,
 
where  

References

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  • 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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