Hadamard ideals and Hadamard matrices with two circulant cores

  title={Hadamard ideals and Hadamard matrices with two circulant cores},
  author={Ilias S. Kotsireas and Christos Koukouvinos and Jennifer Seberry},
  journal={Eur. J. Comb.},
We apply computational algebra methods to the construction of Hadamard matrices with two circulant cores, given by Fletcher, Gysin and Seberry. We introduce the concept of Hadamard ideal, to systematize the application of computational algebra methods for this construction. We use the Hadamard ideal formalism to perform exhaustive search constructions of Hadamard matrices with two circulant cores for the twelve orders 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52. The total number of such… Expand
Heuristic algorithms for Hadamard matrices with two circulant cores
Heuristic algorithms to construct Hadamard matrices with two circulant cores based on local and tabu search and they use information on the geometry of the objective function landscapes to detect when solutions of a special structure exist. Expand
New skew-Hadamard matrices via computational algebra
This paper formalizes three constructions for skew-Hadamard matrices from a Computational Algebra point of view, and shows how to use the doubling construction to construct inequivalent skew- hadamardMatrices of order 2n from skew- Hadamard Matrices ofOrder n. Expand
Gröbner bases and cocyclic Hadamard matrices
An alternative polynomial ideal that also characterizes the set of cocyclic Hadamard matrices over a fixed finite group G of order 4t is described and the complexity of the computation decreases to 2 O ( t ) . 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
Inequivalent Hadamard matrices with buckets
Abstract We give an optimal algorithm to locate all inequivalent Hadamard matrices, given any finite set of Hadamard matrices of the same dimension. When the cardinality of the original set is big,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
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
Hadamardovy matice a jejich využití v kryptografii
This thesis takes interest in Hadamard matrices, their constructions and application in cryptography. Firstly, we introduce basic properties of Hadamard matrices and selected summary of classicalExpand
Hadamard 2-(63, 31, 15) designs invariant under the dihedral group of order 10
It is established that an affine 2-(64,16,5) design of 2-rank 16 that is associated with a Hadamard 2-(63,31,15) design invariant under the dihedral group of order 10 is either isomorphic to the classical design of the points and hyperplanes in AG(3,4), or is one of the two exceptional designs found by Harada, Lam and Tonchev. Expand
A feasibility approach for constructing combinatorial designs of circulant type
This work proposes an optimization approach for constructing various classes of circulant combinatorial designs that can be defined in terms of autocorrelation, and explicitly construct two newcirculant weighing matrices, whose existence was previously marked as unresolved in the most recent version of Strassler’s table. Expand


Hadamard ideals and Hadamard matrices with circulant core
Computational Algebra methods have been used successfully in various problems in many fields of Mathematics. Computational Algebra encompasses a set of powerful algorithms for studying ideals inExpand
Hadamard matrices, Sequences, and Block Designs
One hundred years ago, in 1893, Jacques Hadamard [31] found square matrices of orders 12 and 20, with entries ±1, which had all their rows (and columns) pairwise orthogonal. These matrices, X =Expand
On Hadamard matrices
Abstract Recent advances in the construction of Hadamard matrices have depended on the existence of Baumert-Hall arrays and four (1, −1) matrices A, B, C, D of order m which are of Williamson type,Expand
Hadamard matrices, orthogonal designs and construction algorithms
We discuss algorithms for the construction of Hadamard matrices. We include discussion of construction using Williamson matrices, Legendre pairs and the discret Fourier transform and the twoExpand
An experimental search and new combinatorial designs via a generalisation of cyclotomy
Cyclotomy can be used to construct a variety of combinatorial designs, for example, supplementary difference sets, weighing matrices and T -matrices. These designs may be obtained by using linearExpand
Existence of cyclic Hadamard difference sets and its relation to binary sequences with ideal autocorrelation
The four smallest cases among these 17 cases are examined and the conjecture that all cyclic Hadamard difference sets have parameter v which falls into one of the three types of v is confirmed for all v ≤ 3435. Expand
Application of the discrete Fourier transform to the search for generalised Legendre pairs and Hadamard matrices
It is shown how to construct an Hadamard matrix of order 2£ + 2 from a GL-pair of length f to enable an exhaustive search for GL-pairs for lengths f and partial searches for other f. Expand
Ideals, varieties, and algorithms - an introduction to computational algebraic geometry and commutative algebra (2. ed.)
The algorithmic roots of algebraic object, called a close relationship between ideals, many of polynomial equations in geometric, object called a more than you, for teaching purposes and varieties, and the solutions and reduce even without copy. Expand
Combinatorics: room squares, sum-free sets, Hadamard matrices
Now welcome, the most inspiring book today from a very professional writer in the world, combinatorics room squares sum free sets hadamard matrices. This is the book that many people in the worldExpand
Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3/e (Undergraduate Texts in Mathematics)
Algebraic Geometry is the study of systems of polynomial equations in one or more variables, asking such questions as: Does the The denominator is taking on this, book interested. This book forExpand