• The first unresolved case is order 32 for which I suspect there are over 33,000 Inequivalent Hadamard Matrices.

• There are certainly hundreds and probably thousands of Inequivalent Hadamard matrices for orders 36, 44, 52, .....

• I conjecture that as the power of two increases (eg 32 is 2^5 while 36 is 2^2x9) the number of inequivalent cases increases dramatically.

• I conjecture that the number of Inequivalent Hadamard Matrices of order 36 which are regular (ie have constant row and column sum) is over 100. I conjecture that there are tens of Reqular Inequivalent Hadamard Matrices of order 36 which are not equivalent to a symmetric Reqular Hadamard Matrices of order 36.

# Matrices of Order 16

Marshall Hall's five inequivalent matrices (16H1 , 16H2 , 16H3 , 16H4 , 16H5 )

# Some Constructions for order 20

Three inequivalent matrices (20H01 , 20H02 , 20H03 ). The first is Paley I Construction, the second and third are Tonchev iii and 1v.

# Noburo Ito's 60 inequivalent matrices of order 24

see "Neil Sloane 's Library List". Profiles of inequivalent matrices. Defining sets for inequivalent matrices.

# Kimura's 487 inequivalent matrices of order 28

see "Neil Sloane 's Library List"

For GECP for some of Kimura's Hadamard matrices Gaussian Elimination with Complete Pivoting.

# Some Constructions for order 32

Sylvester Construction (32Syl ), Paley I Construction (32P02 ), Paley II Construction (P12, P13, P14, P15, P16, P17, P18, P19), Marshall Hall Difference Set Construction (32H03 ), W D Wallis Inequivalent (Code 32G05, 32G06, 32G07, 32G08, 32G09, 32G10, 32G11, 32G12, 32G13, 32G14, 32G15).

Also refer to "Neil Sloane's Library List"

# Some Constructions for order 36

Eleven matrices found by Vladimir Tonchev
36H140 , 36H141 , 36H142 , 36H143 , 36H144 , 36H145 , 36H146 , 36H147 , 36H148 , 36H149 , 36H150 ).

179 Further Hadamard matrices of order 36
36H

Bush-type Hadamard matrix of order 36 found by Zvonimir Janko
36J

Regular Hadamard matrices of order 36 found by Jennifer Seberry
36R