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. 

Figures and Tables from this paper

A Concise Guide to Complex Hadamard Matrices

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.

A Path to Hadamard Matrices

Here an algorithm is given to obtain a Hadamard matrix from a matrix of 1s using optimisation techniques on a row-by-row basis.

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

Trades in complex Hadamard matrices

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

Construction, classification and parametrization of complex Hadamard matrices

The intended purpose of this work is to provide the reader with a comprehensive, state-of-the art presentation of the theory of complex Hadamard matrices, or at least report on the very recent

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

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

References

SHOWING 1-10 OF 38 REFERENCES

Classification of Hadamard matrices of order 28

Construction of hadamard matrices of order 28

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

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].

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

Invariant factors of combinatorial matrices

The Smith normal forms of an Hadamard matrix of order 4m (m square-free), and of the incidence matrix of a (ν, k, λ) configuration (n=k−λ square-free (n, λ)=1), are determined.