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
Pages (from-to)627-642
Number of pages16
JournalJournal of Combinatorial Designs
Volume27
Issue number11
DOIs
Publication statusPublished - 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",
doi = "10.1002/jcd.21664",
language = "English",
volume = "27",
pages = "627--642",
journal = "Journal of Combinatorial Designs",
issn = "1063-8539",
publisher = "John Wiley & Sons",
number = "11",

}

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

In: Journal of Combinatorial Designs, Vol. 27, No. 11, 2019, p. 627-642.

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

Y1 - 2019

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

UR - http://www.scopus.com/inward/record.url?scp=85071339467&partnerID=8YFLogxK

U2 - 10.1002/jcd.21664

DO - 10.1002/jcd.21664

M3 - Article

VL - 27

SP - 627

EP - 642

JO - Journal of Combinatorial Designs

JF - Journal of Combinatorial Designs

SN - 1063-8539

IS - 11

ER -