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Research - Hadamard Matrices

- The Hadamard matrix - as usual, -1 is represented as -.
- The automorphism group as a permutation group. For a Hadamard matrix of
order
n , the group acts on2n points, the firstn of which are the rows of the matrix, and th remainingn the negations of those rows (in the same order). Thus a given element of this group uniquely determines a sequence of permutations and negations of rows. Since a Hadamard matrix is invertible, this uniquely determines an action on the columns which makes this into an automorphism. - Data on the regular subgroups of the automorphism group. These correspond
to group development schemes for the expanded matrix and cocyclic development
schemes for the Hadamard matrix itself. Only the isomorphism types of the
groups are given, in the form of their position in the small groups library.
These may be visualised as short exact sequences of the type

$1\; \to \; C\_2\; \to \; E\; \to \; G\; \to \; 1$ where G appears in the first column and E in the second. Such sequences may appear repeatedly, in this case they are not cohomologically equivalent. For reasons of space, the cocycle is not given. For f urther details see my Masters Thesis.

- Orders less than 16
- Order 16
- Order 20
- Order 24 (In the Magma database, these matrices are numbered 1,3,4,5,6,9,10,20,22,24,27,47,51,56,57,60.)
- Order 28 (In the Magma database these matrices are numbered 1,10,30,109,250,278.)
- Order 32 (Details of 70 matrices are contained in the file. The remaining thirty are transpose equivalent to ones listed, and so have identical cocyclic development properties. The following forty matrices are transpose-self-equivalent: 1,6,7,12,15,22-27,29-35,38,41,44,49-52,55-60,62-70.)
- Order 36 (Details of 26 matrices are contained in the file. The other 9 are transpose equivalent to ones listed, and so have identical cocyclic development properties. The following 17 matrices are self-transpoe-equivalent: 1,7,9,10,12-14,16,18-26.)
### Data Files

It may be convenient for some people to have access to these matrices in a form that can be immediately input into e.g. MAGMA. They are given below. Once a file is loaded in MAGMA, the command Matrix(Had32[i]) gives the i^th cocyclic Hadamard matrix of order 32. If you find these files, or any of the the information on this page of use, please send me a short e-mail about it.

School of Mathematics, Statistics and Applied Mathematics

National University of Ireland, Galway, University Road, Galway, Ireland.

Phone: +353 (0)91 492332 (direct) , +353 (0)91 524411 x2332 (switchboard)

Fax: +353 (0)91 494542 Email: Mary.Kelly@nuigalway.ie

This page was last updated Thursday, June 18

National University of Ireland, Galway, University Road, Galway, Ireland.

Phone: +353 (0)91 492332 (direct) , +353 (0)91 524411 x2332 (switchboard)

Fax: +353 (0)91 494542 Email: Mary.Kelly@nuigalway.ie

This page was last updated Thursday, June 18