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,
Existence
If p is prime then can exist only for with integer m and it is conjectured they exist for all such cases with . In general, the problem of finding all sets such that the Butson - type matrices exist, remains open.
Examples
- 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
- 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).
External links
- Complex Hadamard Matrices of Butson type - a catalogue, by Wojciech Bruzda, Wojciech Tadej and Karol Życzkowski, retrieved October 24, 2006
This article is issued from Wikipedia - version of the 3/21/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.