SKEW-HADAMARD MATRICES



© Nickolay Balonin and Jennifer Seberry, 1.09.2014

Skew-Hadamard matrix catalogue and on-line algorithms


Definition. The Hadamard matrix H is nxn ±1-matrix H'H=nI. For p=n–1 is prime, in the cases n=4, 8, 12, 24, 32, 44, 48, 60, 68, 72, 80, 84 ..., there is one border one circulant construction with Legendre symbols. There are also two- and four-circulant constructions.

H=
H
A
 –A 
H
;
H=
A
B
C
D
 –B 
A
D
 –C 
 –C 
 –D 
A
B
 –D 
C
 –B 
A


For even order 16 we use a two circulant version: H is a skew Hadamard matrix while A is a symmetric Hadamard matrix (i.e. AT=A). We use some non symmetry blocks of a fractal construction, based on an idea of chains, especially for orders n=2t, t is integer, 2, 4, 8, 32, 64, 128, 512, .. presented below.

FOUR CIRCULANT SKEW-HADAMARD MATRICES


The skew-Hadamard matrix H can be calculated by Williamson-type A, B, C, D matrices, called good matrices, where AAT+BBT+CCT+DDT=4mI, A is skew-circulant and B, C, D are symmetric circulant matrices that are placed in the Seberry-Williamson array as back-circulant matrices, symmetric about the back diagonal. Skew Hadamard matrices are known to exist for for all n<188 with n divisible by 4 (look Warren D. Smith, August 2006), below placed matrices found by Christos Koukouvinos (mostly).




Matrices H4 and H12



Matrices H20 and H28



Matrices H36 and H44



Matrices H52 and H60



Matrices H68 and H76



Matrices H84 and H92



Matrices H100!

THE FIRST COMPUTER RESEARHES: 1961




The pleasure to find new matrix: Hadamard 92 moment 1961


From left to right, Solomon Golomb (Assistant Chief of the Communications Systems Research Section), Leonard Baumert (a postdoc student at Caltech), and Marshall Hall, Jr. (Caltech mathematics professor) hold a framed representation of the matrix.

In 1961, mathematicians from NASA’s Jet Propulsion Laboratory and Caltech worked together to construct a Hadamard Matrix containing 92 rows and columns, with combinations of positive and negative signs. In a Hadamard Matrix, if you placed all the potential rows or columns next to each other, half of the adjacent cells would be the same sign, and half would be the opposite sign. This mathematical problem had been studied since about 1893, but the solution to the 92 by 92 matrix was unproven until 1961 because it required extensive computation. The team used JPL’s IBM 7090 computer, programmed by Baumert, to perform the computations.

THE FIRST COMPUTER RESEARCHES: 1971


There is a skew-Hadamard matrix of order 92 (Jennifer Seberry, 1971). Previously the smallest order for which a skew-Hadamard matrix was not known was 92. The existence of any Hadamard matrix of order 92 was unknown until 1962 (see photo with Williamson matrix, order 92).

ON LINE ALGORITHM. A SAMPLE OF H44


GET MORE COLOURS 44x44


Jennifer Seberry Wallis, A skew-Hadamard matrix of order 92, Bulletin of the Australian Mathematical Society, 5, (1971), 203-204.



CATALOGUE | SKEW PROPUS-HADAMARD MATRICES


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