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# Number of equivalent Hadamard matrices

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### budi...@gmail.com

Aug 14, 2016, 8:14:30 AM8/14/16
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Hadamard matrices exist for most sizes that are multiples of 4. A great
deal of research was done to find for which matrix size these Hadamard
matrices exist and to determine the number of non-equivalent Hadamard
matrices. To cite Wikipedia: "Two Hadamard matrices are considered
equivalent if one can be obtained from the other by negating rows or
columns, or by interchanging rows or columns. Up to equivalence, there
is a unique Hadamard matrix of orders 1, 2, 4, 8, and 12. There are 5
inequivalent matrices of order 16, 3 of order 20, 60 of order 24, and
487 of order 28. Millions of inequivalent matrices are known for orders
32, 36, and 40."

However what I am interested in is to find how many equivalent matrices
exist for a given inequivalent matrix. The maximum number of equivalent
matrices for matrix size M x M would be:

2^M x 2^M x M! x M!

However if you make all sign changes and row and column permutations
you end up with many of repetition. I am looking for some research
which would determine the number of unique equivalent matrices. Or
which would identify the source of repetitions.
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