Constructing cocyclic Hadamard matrices of order 4p

Research output: Contribution to journalArticleResearchpeer-review

Abstract

Cocyclic Hadamard matrices (CHMs) were introduced by de Launey and Horadam as a class of Hadamard matrices (HMs) with interesting algebraic properties. Ó Catháin and Röder described a classification algorithm for CHMs of order 4n based on relative difference sets in groups of order 8n; this led to the classification of all CHMs of order at most 36. On the basis of work of de Launey and Flannery, we describe a classification algorithm for CHMs of order 4p with pa prime; we prove refined structure results and provide a classification for p ≤ 13. Our analysis shows that every CHM of order 4p with p ≡ 1 is equivalent to an HM with one of five distinct block structures, including Williamson-type and (transposed) Ito matrices. If p ≡ 3, then every CHM of order 4p is equivalent to a Williamson-type or (transposed) Ito matrix.

Original languageEnglish
Number of pages16
JournalJournal of Combinatorial Designs
DOIs
Publication statusAccepted/In press - 27 Aug 2019

Keywords

  • cocyclic development
  • Hadamard matrix
  • Ito type
  • Williamson type

Cite this

@article{262d6052f77041c1abbd612144c129ec,
title = "Constructing cocyclic Hadamard matrices of order 4p",
abstract = "Cocyclic Hadamard matrices (CHMs) were introduced by de Launey and Horadam as a class of Hadamard matrices (HMs) with interesting algebraic properties. {\'O} Cath{\'a}in and R{\"o}der described a classification algorithm for CHMs of order 4n based on relative difference sets in groups of order 8n; this led to the classification of all CHMs of order at most 36. On the basis of work of de Launey and Flannery, we describe a classification algorithm for CHMs of order 4p with pa prime; we prove refined structure results and provide a classification for p ≤ 13. Our analysis shows that every CHM of order 4p with p ≡ 1 is equivalent to an HM with one of five distinct block structures, including Williamson-type and (transposed) Ito matrices. If p ≡ 3, then every CHM of order 4p is equivalent to a Williamson-type or (transposed) Ito matrix.",
keywords = "cocyclic development, Hadamard matrix, Ito type, Williamson type",
author = "{Barrera Acevedo}, Santiago and Cathain, {Padraig O.} and Heiko Dietrich",
year = "2019",
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language = "English",
journal = "Journal of Combinatorial Designs",
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publisher = "John Wiley & Sons",

}

Constructing cocyclic Hadamard matrices of order 4p. / Barrera Acevedo, Santiago; Cathain, Padraig O.; Dietrich, Heiko.

In: Journal of Combinatorial Designs, 27.08.2019.

Research output: Contribution to journalArticleResearchpeer-review

TY - JOUR

T1 - Constructing cocyclic Hadamard matrices of order 4p

AU - Barrera Acevedo, Santiago

AU - Cathain, Padraig O.

AU - Dietrich, Heiko

PY - 2019/8/27

Y1 - 2019/8/27

N2 - Cocyclic Hadamard matrices (CHMs) were introduced by de Launey and Horadam as a class of Hadamard matrices (HMs) with interesting algebraic properties. Ó Catháin and Röder described a classification algorithm for CHMs of order 4n based on relative difference sets in groups of order 8n; this led to the classification of all CHMs of order at most 36. On the basis of work of de Launey and Flannery, we describe a classification algorithm for CHMs of order 4p with pa prime; we prove refined structure results and provide a classification for p ≤ 13. Our analysis shows that every CHM of order 4p with p ≡ 1 is equivalent to an HM with one of five distinct block structures, including Williamson-type and (transposed) Ito matrices. If p ≡ 3, then every CHM of order 4p is equivalent to a Williamson-type or (transposed) Ito matrix.

AB - Cocyclic Hadamard matrices (CHMs) were introduced by de Launey and Horadam as a class of Hadamard matrices (HMs) with interesting algebraic properties. Ó Catháin and Röder described a classification algorithm for CHMs of order 4n based on relative difference sets in groups of order 8n; this led to the classification of all CHMs of order at most 36. On the basis of work of de Launey and Flannery, we describe a classification algorithm for CHMs of order 4p with pa prime; we prove refined structure results and provide a classification for p ≤ 13. Our analysis shows that every CHM of order 4p with p ≡ 1 is equivalent to an HM with one of five distinct block structures, including Williamson-type and (transposed) Ito matrices. If p ≡ 3, then every CHM of order 4p is equivalent to a Williamson-type or (transposed) Ito matrix.

KW - cocyclic development

KW - Hadamard matrix

KW - Ito type

KW - Williamson type

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