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gap-packages / guava

GAP package guava - computations relative to error-correcting codes
https://gap-packages.github.io/guava
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HadamardMat improvements #14

Open osj1961 opened 8 years ago

osj1961 commented 8 years ago

There is a paper, DISCOVERY OF AN HADAMARD MATRIX OF ORDER 92 by LEONARD BAUMERT, S. W. GOLOMB AND MARSHALL HALL, JR. describing a construction using a 4x4 block form

[  A  B  C  D ]
[ -B  A -D  C ]
[ -C  D  A -B ]
[ -D -C  B  A ]

where A,B,C & D are 23x23 (symmetric) circulant matrices.

They claim the method can be generalized to cover 92, 116, 156, and 188 (also 172 which was originally done by J. Williamson using "a special automorphism of order 3") The amount of computation involved was apparently prohibitive in 1961. The technique may have to do with finding multiple difference sets in GF(m/4).

osj1961 commented 8 years ago

Found another paper, Hadamard Matrices of the Williamson Type By L. D. Baumert and Marshall Hall, Jr. w/ more details. The first rows of the circulant matrices correspond to solutions of x^2+y^2+z^2+w^2=4t in odd integers but written in terms of polynomials in $\omega_i$ which is left undefined...

osj1961 commented 8 years ago

Neil Sloan maintains a webpage about Hadamard matrices: http://neilsloane.com/hadamard/index.html where in addition to Williamson type he also mentions Turyn type. There are several references...