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,

:: $(H\_\{jk\})^q=1\; \{quad\; m\; for\; quad\}\; j,k=1,2,dots,N.$

Existence

If "p" is prime then $H(p,N)$ can existonly for $N\; =\; mp$ with integer "m" andit 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

*$H(2,N)$ contains real Hadamard matrices of size "N",

*$H(4,N)$ contains Hadamard matrices composed of $pm\; 1,\; pm\; i$ - such matrices were called by Turyn, complex Hadamard matrices.

* in the limit $q\; o\; infty$ one can approximate all complex Hadamard matrices.

*Fourier matrices $[F\_N]\; \_\{jk\}:=\; exp\; [(2pi\; i(j\; -\; 1)(k\; -\; 1)\; /\; N]\; \{quad\; m\; for\; quad\}\; j,k=1,2,dots,N$belong to the Butson-type,

:: $F\_N\; in\; H(N,N)$,

: while

:: $F\_N\; otimes\; F\_N\; in\; H(N,N^2)$,

:: $F\_N\; otimes\; F\_Notimes\; F\_N\; in\; H(N,N^3)$.

::

:: , where $z\; =exp(2pi\; i/3)$.

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, 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

[http://chaos.if.uj.edu.pl/~karol/hadamard/index.php?I=0200 Complex Hadamard Matrices of Butson type - a catalogue] , by Wojciech Bruzda, Wojciech Tadej and Karol Życzkowski, retrieved October 24, 2006