# Hadamard matrices

A *Hadamard matrix* of order *n* is an *n* x *n*
matrix of +1 and -1 whose rows (and therefore columns) are
orthogonal. The famous Hadamard conjecture is that a least
one Hadamard matrix exists for each order that is a multiple of 4.

In the following files there is one Hadamard matrix per line.
The line consists of *n* hexadecimal numbers, one per row
of the matrix. To obtain the actual rows, write the numbers in
binary and change all zeros to -1.

Two Hadamard matrices are *equivalent* if one can be
obtained from the other by permuting rows, permuting columns,
and negating rows and columns.

The Hadamard matrices of order 32 were determined by Hadi
Kharaghani and Behruz Tayfeh-Rezaie.

## Equivalence classes of Hadamard matrices

order 4 (1)

order 8 (1)

order 12 (1)

order 16 (5)

order 20 (3)

order 24 (60)

order 28 (487)

order 32 (gzipped) (13710027)

For order 32, we also provide two further files. The first
contains representatives of the extended equivalence classes
when the transpose is regarded as equivalent to the original.
The second contains representatives of the equivalence classes
of matrices equivalent to their own transposes.

extended classes (gzipped) (6857010)

transpose-equivalent classes (gzipped) (3993)

Page Master: Brendan McKay,
brendan.mckay@anu.edu.au

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