Ronan Egan
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View article: Centraliser algebras of monomial representations and applications in combinatorics
Centraliser algebras of monomial representations and applications in combinatorics Open
Centraliser algebras of monomial representations of finite groups may be constructed and studied using methods similar to those employed in the study of permutation groups. Guided by results of D. G. Higman and others, we give an explicit …
View article: A generalisation of bent vectors for Butson Hadamard matrices
A generalisation of bent vectors for Butson Hadamard matrices Open
An $n\times n$ complex matrix $M$ with entries in the $k^{\textrm{th}}$ roots of unity which satisfies $MM^{\ast} = nI_{n}$ is called a Butson Hadamard matrix. While a matrix with entries in the $k^{\textrm{th}}$ roots typically does not h…
View article: COMPLEX WEIGHING MATRICES AND QUATERNARY CODES
COMPLEX WEIGHING MATRICES AND QUATERNARY CODES Open
Weighing matrices with entries in the complex cubic and sextic roots of unity are employed to construct Hermitian self-dual codes and Hermitian linear complementary dual codes over the finite field $\mathrm {GF}(4).$ The parameters of thes…
View article: Centraliser algebras of monomial representations and applications in combinatorics
Centraliser algebras of monomial representations and applications in combinatorics Open
Centraliser algebras of monomial representations of finite groups may be constructed and studied using methods similar to those employed in the study of permutation groups. Guided by results of D. G. Higman and others, we give an explicit …
View article: Butson Hadamard matrices, bent sequences, and spherical codes
Butson Hadamard matrices, bent sequences, and spherical codes Open
We explore a notion of bent sequence attached to the data consisting of an Hadamard matrix of order $n$ defined over the complex $q^{th}$ roots of unity, an eigenvalue of that matrix, and a Galois automorphism from the cyclotomic field of …
View article: A survey of complex generalized weighing matrices and a construction of quantum error-correcting codes
A survey of complex generalized weighing matrices and a construction of quantum error-correcting codes Open
Some combinatorial designs, such as Hadamard matrices, have been extensively researched and are familiar to readers across the spectrum of Science and Engineering. They arise in diverse fields such as cryptography, communication theory, an…
View article: Generalized partially bent functions, generalized perfect arrays, and cocyclic Butson matrices
Generalized partially bent functions, generalized perfect arrays, and cocyclic Butson matrices Open
In a recent survey, Schmidt compiled equivalences between generalized bent functions, group invariant Butson Hadamard matrices, and abelian splitting relative difference sets. We establish a broader network of equivalences by considering B…
View article: Generalized partially bent functions, generalized perfect arrays and cocyclic Butson matrices
Generalized partially bent functions, generalized perfect arrays and cocyclic Butson matrices Open
In a recent survey, Schmidt compiled equivalences between generalized bent functions, group invariant Butson Hadamard matrices, and abelian splitting relative difference sets.We establish a broader network of equivalencesby considering But…
View article: Generalized partially bent functions, generalized perfect arrays and cocyclic Butson matrices
Generalized partially bent functions, generalized perfect arrays and cocyclic Butson matrices Open
In a recent survey, Schmidt compiled equivalences between generalized bent functions, group invariant Butson Hadamard matrices, and abelian splitting relative difference sets. We establish a broader network of equivalences by considering B…
View article: A Survey of the Hadamard Maximal Determinant Problem
A Survey of the Hadamard Maximal Determinant Problem Open
In a celebrated paper of 1893, Hadamard established the maximal determinant theorem, which establishes an upper bound on the determinant of a matrix with complex entries of norm at most 1. His paper concludes with the suggestion that mathe…
View article: Butson full propelinear codes
Butson full propelinear codes Open
In this paper we study Butson Hadamard matrices, and codes over finite rings coming from these matrices in logarithmic form, called BH-codes. We introduce a new morphism of Butson Hadamard matrices through a generalized Gray map on the mat…
View article: Self-orthogonal codes from orbit matrices of Seidel and Laplacian matrices of strongly regular graphs
Self-orthogonal codes from orbit matrices of Seidel and Laplacian matrices of strongly regular graphs Open
In this paper we introduce the notion of orbit matrices of integer matrices such as Seidel and Laplacian matrices of some strongly regular graphs with respect to their permutation automorphism groups. We further show that under certain con…
View article: Spectra of Hadamard matrices
Spectra of Hadamard matrices Open
A Butson Hadamard matrix $H$ has entries in the kth roots of unity, and satisfies the matrix equation $HH^{\ast} = nI_{n}$. We write $\mathrm{BH}(n, k)$ for the set of such matrices. A complete morphism of Butson matrices is a map $\mathrm…