On the classification of Hadamard matrices of order 32

This chapter introduces some of the basic ingredients of cohomological group theory and describes datatypes and algorithms for implementing them on a computer. These are illustrated using computer examples involving: explicit cocycles, classification of abelian and nonabelian group extensions, crossed modules, crossed extensions, five-term exact sequences, Hopf’s formula, Bogomolov multipliers, relative central extensions, nonabelian tensor products of groups, and cocyclic Hadamard matrices.

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Towards a Classification of 6 × 6 Complex Hadamard Matrices

Open Systems & Information Dynamics ◽

10.1142/s1230161208000092 ◽

2008 ◽

Vol 15 (02) ◽

pp. 93-108 ◽

Cited By ~ 14

Author(s):

Máté Matolcsi ◽

Ferenc Szöllősi

Keyword(s):

Information Theory ◽

Quantum Information ◽

Hadamard Matrices ◽

Quantum Information Theory ◽

Higher Order ◽

Complete Characterization ◽

The Past ◽

Symmetric Complex ◽

Fourier Matrix

Complex Hadamard matrices have received considerable attention in the past few years due to their application in quantum information theory. While a complete characterization currently available [5] is only up to order 5, several new constructions of higher order matrices have appeared recently [4, 12, 2, 7, 11]. In particular, the classification of self-adjoint complex Hadamard matrices of order 6 was completed by Beuachamp and Nicoara in [2], providing a previously unknown non-affine one-parameter orbit. In this paper we classify all dephased, symmetric complex Hadamard matrices with real diagonal of order 6. Furthermore, relaxing the condition on the diagonal entries we obtain a new non-affine one-parameter orbit connecting the Fourier matrix F6 and Diţă's matrix D6. This answers a recent question of Bengtsson et al. [3].

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Construction and classification of group invariant Butson Hadamard matrices

10.32657/10356/145710 ◽

2020 ◽

Author(s):

◽

Dai Quan Wong

Keyword(s):

Hadamard Matrices ◽

Group Invariant

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Classification of Generalized Hadamard Matrices $H(6,3)$ and Quaternary Hermitian Self-Dual Codes of Length 18

The Electronic Journal of Combinatorics ◽

10.37236/443 ◽

2010 ◽

Vol 17 (1) ◽

Cited By ~ 3

Author(s):

Masaaki Harada ◽

Clement Lam ◽

Akihiro Munemasa ◽

Vladimir D. Tonchev

Keyword(s):

Hadamard Matrices ◽

Dual Codes ◽

Transversal Designs

All generalized Hadamard matrices of order 18 over a group of order 3, $H(6,3)$, are enumerated in two different ways: once, as class regular symmetric $(6,3)$-nets, or symmetric transversal designs on 54 points and 54 blocks with a group of order 3 acting semi-regularly on points and blocks, and secondly, as collections of full weight vectors in quaternary Hermitian self-dual codes of length 18. The second enumeration is based on the classification of Hermitian self-dual $[18,9]$ codes over $GF(4)$, completed in this paper. It is shown that up to monomial equivalence, there are 85 generalized Hadamard matrices $H(6,3)$, and 245 inequivalent Hermitian self-dual codes of length 18 over $GF(4)$.

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