Switching Operations for Hadamard Matrices

@article{Orrick2008SwitchingOF,
  title={Switching Operations for Hadamard Matrices},
  author={William P. Orrick},
  journal={SIAM J. Discret. Math.},
  year={2008},
  volume={22},
  pages={31-50}
}
  • W. Orrick
  • Published 25 July 2005
  • Mathematics
  • SIAM J. Discret. Math.
We define several operations that switch substructures of Hadamard matrices, thereby producing new, generally inequivalent, Hadamard matrices. These operations have application to the enumeration and classification of Hadamard matrices. To illustrate their power, we use them to greatly improve the lower bounds on the number of equivalence classes of Hadamard matrices in orders 32 and 36 to 3,578,006 and 18,292,717. 

Tables from this paper

A Concise Guide to Complex Hadamard Matrices
TLDR
Basic properties of complex Hadamard matrices are reviewed and a catalogue of inequivalent cases known for the dimensions N = 2, 16, 12, 14 and 16 are presented.
Trades in complex Hadamard matrices Padraig Ó Catháin
A trade in a complex Hadamard matrix is a set of entries which can be changed to obtain a different complex Hadamard matrix. We show that in a real Hadamard matrix of order n all trades contain at
The quaternary complex Hadamard matrices of orders 10, 12, and 14
On D4t ‐Cocyclic Hadamard Matrices
In this paper, we describe some necessary and sufficient conditions for a set of coboundaries to yield a cocyclic Hadamard matrix over the dihedral group D4t . Using this characterization, new
Switching codes and designs
Equivalences of Zt×Z22-cocyclic Hadamard matrices
One of the most promising structural approaches to resolving the Hadamard Conjecture uses the family of cocyclic matrices over Zt × Z 2 2. Two types of equivalence relations for classifying cocyclic
Group actions on Hadamard matrices
Faculty of Arts Mathematics Department Master of Literature by Padraig Ó Catháin Hadamard matrices are an important item of study in combinatorial design theory. In this thesis, we explore the theory
Robust Hadamard Matrices, Unistochastic Rays in Birkhoff Polytope and Equi-Entangled Bases in Composite Spaces
TLDR
It is shown that such a matrix of order n exists, if there exists a skew Hadamard matrix or a symmetric conference matrix of this size, and it is demonstrated that a bistochastic matrix B located at any ray of the Birkhoff polytope, (which joins the center of this body with any permutation matrix), is unistochastics.
Diagram versus bundle equivalence for ℤt × ℤ22-cocyclic Hadamard matrices
TLDR
It is shown the group Bund(t) generated by bundle equivalence operations is isomorphic to a subgroup of index 2 in the group Diag(t), and that Diag (t) = where T is the geometric translation of matrix transposition.
...
...

References

SHOWING 1-10 OF 40 REFERENCES
Classification of Hadamard matrices of order 28
Construction of hadamard matrices of order 28
TLDR
In this paper only 20 matrices are listed and they are constructed from Hadamard matrices with Hall sets of order 28 containing 13 matrices in [7] and [8].
Equivalence of Hadamard matrices
Supposem is a square-free odd integer, andA andB are any two Hadamard matrices of order 4m. We will show thatA andB are equivalent over the integers (that is,B can be obtained fromA using elementary
Integral equivalence of hadamard matrices
SupposeA is a non-singular matrix with entries 0 and 1, the zero and identity elements of a Euclidean domain. We obtain a “best-possible” lower bound for the number of equivalence invariants ofA
Hadamard equivalence via graph isomorphism
New Hadamard matrix of order 24
TLDR
The classification of Hadamard matrices of order 24 is completed by this paper and Ito-Leon-Longyear [3] and this matrix must be appear in [11].
Some Hadamard matrices of order 32 and their binary codes
TLDR
It is demonstrated that every extremal doubly‐even self‐dual [32,16,8] code can be constructed from some binary Hadamard matrix of order 32.
On Maximal Weights of Hadamard Matrices
A Block Negacyclic Bush-Type Hadamard Matrix and Two Strongly Regular Graphs
A block negacyclic Bush-type Hadamard matrix of order 36 is used in a symmetric BGW(26, 25, 24) with zero diagonal over a cyclic group of order 12 to construct a twin strongly regular graph with
Classification of 3-(24, 12, 5) Designs and 24-Dimensional Hadamard Matrices
...
...