The cocyclic Hadamard matrices of order less than 40

  title={The cocyclic Hadamard matrices of order less than 40},
  author={Padraig {\'O} Cath{\'a}in and Marc R{\"o}der},
  journal={Designs, Codes and Cryptography},
In this paper all cocyclic Hadamard matrices of order less than 40 are classified. That is, all such Hadamard matrices are explicitly constructed, up to Hadamard equivalence. This represents a significant extension and completion of work by de Launey and Ito. The theory of cocyclic development is discussed, and an algorithm for determining whether a given Hadamard matrix is cocyclic is described. Since all Hadamard matrices of order at most 28 have been classified, this algorithm suffices to… Expand
Cocyclic two-circulant core Hadamard matrices
The two-circulant core (TCC) construction for Hadamard matrices uses two sequences with almost perfect autocorrelation to construct a Hadamard matrix. A research problem of K. Horadam asks whetherExpand
Constructing cocyclic Hadamard matrices of order 4p
Cocyclic Hadamard matrices (CHMs) were introduced by de Launey and Horadam as a class of Hadamard matrices with interesting algebraic properties. \'O Cath\'ain and R\"oder described a classificationExpand
Cocyclic Hadamard matrices over Latin rectangles
It is proved that every Latin square over which a Hadamard matrix is cocyclic must be the multiplication table of a loop (not necessarily associative) and the existence of cocYclic Hadamards matrices over non-associative loops of order 2 t + 3 , for all positive integer t > 0 . Expand
On Cocyclic Hadamard Matrices over Goethals-Seidel Loops
About twenty-five years ago, Horadam and de Launey introduced the cocyclic development of designs, from which the notion of cocyclic Hadamard matrices developed over a group was readily derived. MuchExpand
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 cocyclicExpand
Difference sets and doubly transitive actions on Hadamard matrices
This work classify all difference sets which give rise to Hadamard matrices with non-affine doubly transitive automorphism group, and uncovers a new triply infinite family of skew-Hadamard difference sets in non-abelian p-groups with no exponent restriction. Expand
Hadamard Matrices with Cocyclic Core
Since Horadam and de Launey introduced the cocyclic framework on combinatorial designs in the 1990s, it has revealed itself as a powerful technique for looking for (cocyclic) Hadamard matrices. TenExpand
On twin prime power Hadamard matrices
This work answers a research problem posed by K.J. Horadam, and exhibits the first known infinite family of Hadamard matrices which are not cocyclic, by showing that the action of the automorphism group of a hadamard matrix developed from a difference set induces a 2-transitive action on the rows of the matrix. Expand
The maximal determinant of cocyclic (-1,1)-matrices over D2t
Abstract Cocyclic construction has been successfully used for Hadamard matrices of order n. These ( - 1 , 1 ) -matrices satisfy that HH T = H T H = nI and give the solution to the maximal determinantExpand
Topics in cocyclic development of pairwise combinatorial designs
This is an abstract of the PhD thesis Topics in Cocyclic Development of Pairwise Combinatorial Designs written by Ronan Egan under the supervision of Dane Flannery at the School of Mathematics,Expand


Cocyclic Hadamard matrices and difference sets
It is proved that the existence of a cocyclic Hadamard matrix of order 4t is equivalent to the existenceof a normal relative difference set with parameters (4t,2,4T,2t, 2t) . Expand
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 theoryExpand
Cocyclic Development of Designs
AbstractWe present the basic theory of cocyclic development of designs, in which group development over a finite group G is modified by the action of a cocycle defined on G × G. Negacyclic andExpand
Cocyclic Hadamard Matrices and Hadamard Groups Are Equivalent
Abstract In this paper, we prove that the concepts of cocyclic Hadamard matrix and Hadamard group are equivalent. A general procedure for constructing Hadamard groups and classifying such groups onExpand
Switching Operations for Hadamard Matrices
  • W. Orrick
  • Mathematics, Computer Science
  • SIAM J. Discret. Math.
  • 2008
These operations are used 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. Expand
Quasiregular Projective Panes of Order 16 -- A Computational Approach
This thesis discusses methods for the classification of finite projective planes via exhaustive search. In the main part the author classifies all projective planes of order 16 admitting a largeExpand
A Hadamard matrix of order 428
Four Turyn type sequences of lengths 36, 36, 36, 35 are found by a computer search. These sequences give new base sequences of lengths 71, 71, 36, 36 and are used to generate a number of newExpand
Classification of hadamard matrices of order 24 and 28
  • E. Spence
  • Computer Science, Mathematics
  • Discret. Math.
  • 1995
Abstract The purpose of this paper is to offer an independent verification of recent computer results of Kimura [2,3,4,5] on the classification of Hadamard matrices of orders 24 and 28.
Hadamard Matrices and Their Applications
Construction of classic Hadamard matrices.- Construction of generalized Hadamard matrices.- Application of Hadamard matrices.
On Hadamard Groups
Abstract Hadamard groups are introduced and some of their basic properties are established.