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Title: Math/Combinatorics/Design Theory - Ted Spence's Home Page Classification of Hadamard matrices and designs.
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Ted Spence's Home Page My research interests at the present time are in the area ofclassification of combinatorial designs of various different sorts.In some cases I have been able to classify the designs completelyand where this has been possible I have stored the designs on disc.They can be accessed via the Table below. These files will be updatedat intervals, as I find the time. The `smallest' parameter set of a symmetric design whose existence isstill in doubt is (81,16,3). One line of attack in constructing such a design might be to assume that it has a fixed-point freeautomorphism of order 3. However, the number of possible tacticaldecompositions is extremely large, and so another assumption would probably have to be made. One, suggested by Vladimir Tonchev, was thatthe tactical decompostion should come from a skew-hadamard matrix oforder 28 in the following way. Normalise the skew-hadamard matrix,delete the first row and column and change all the -1's to zeros andthen replace the zeros on the diagonal by 3's. The resulting matrix is a possible tactical decomposition for the given design and automorphism. Recent work of my own entailed the classification of all Hadamard matrices of order 28 and among these I had isolated theskew ones. From these I was able to determine all possible non-isomorphic tactical decompositions constructed in the above manner.These numbers were further reduced by eliminating those decompositionsthat were isomorphic to their transposes (corresponding to the dualdesign). The final count was 411 tactical decompositions to be examined.After a fairly lengthy computer investigation it was finally shown that no (81,16,3) design could be constructed with the above assumptions. The reason that I announce this here is to save a fellowresearcher from spending some time on what has turned out to be afruitless task. The following link Symmetric Designs in Combinatorics will give the reader a good list of books and journals for further study on the subject of symmetric designs If you do download any of my files, it would be appreciated ifyou would e-mail me a message to let me know that you have done so:ted@maths.gla.ac.uk List of Publications 2-designsHadamard matrices Two-graphs Steiner S(2,4,25) Regular graphs with four eigenvalues Strongly regular graphs on at most 64 verticesSymmetric designs with a non-null polarity Address:Department of MathematicsUniversity of GlasgowGlasgow G12 8QQSCOTLAND Phone:+44 (0)141 330-4356 Fax: +44 (0)141 330-4111
 

Classification

of

Hadamard

matrices

and

designs.

http://www.maths.gla.ac.uk/~es/

Ted Spence's Home Page 2008 November

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Classification of Hadamard matrices and designs.

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