New infinite families of Williamson Hadamard matrices

Research output: Contribution to journalArticleResearchpeer-review

Abstract

Due to the Hadamard Conjecture, Williamson matrices (WM) and Williamson type matrices (WTM) of order 4n have been primarily investigated for odd n. Several constructions for this case have been introduced, leading to finite and infinite families of WMs and WTMs. The aim of this paper is to present new families of WMs and WTMs with blocks of even order. Let q and r be prime powers congruent to 1 modulo 4. There are WMs of order 4a(q + 1) for every a ∈ {1, 11, 17, 23, 29, 33, 39, 43}. If gcd(q +1, r +1) = 2, then there is a WM of order 2(q +1)(r+1). There are WMs of order 2b(q + 1) and WTMs with circulant blocks of order 2c(q + 1) for every b ∈ {2, . . . , 7} and c ∈ {5, 6}. We prove these results and more by exploiting a recently established correspondence between perfect quaternionic sequences and relative difference sets.
Original languageEnglish
Pages (from-to)207-219
Number of pages13
JournalAustralasian Journal of Combinatorics
Volume73
Issue number1
Publication statusPublished - Jan 2019

Cite this

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title = "New infinite families of Williamson Hadamard matrices",
abstract = "Due to the Hadamard Conjecture, Williamson matrices (WM) and Williamson type matrices (WTM) of order 4n have been primarily investigated for odd n. Several constructions for this case have been introduced, leading to finite and infinite families of WMs and WTMs. The aim of this paper is to present new families of WMs and WTMs with blocks of even order. Let q and r be prime powers congruent to 1 modulo 4. There are WMs of order 4a(q + 1) for every a ∈ {1, 11, 17, 23, 29, 33, 39, 43}. If gcd(q +1, r +1) = 2, then there is a WM of order 2(q +1)(r+1). There are WMs of order 2b(q + 1) and WTMs with circulant blocks of order 2c(q + 1) for every b ∈ {2, . . . , 7} and c ∈ {5, 6}. We prove these results and more by exploiting a recently established correspondence between perfect quaternionic sequences and relative difference sets.",
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language = "English",
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}

New infinite families of Williamson Hadamard matrices. / Barrera Acevedo, Santiago; Dietrich, Heiko.

In: Australasian Journal of Combinatorics, Vol. 73, No. 1, 01.2019, p. 207-219.

Research output: Contribution to journalArticleResearchpeer-review

TY - JOUR

T1 - New infinite families of Williamson Hadamard matrices

AU - Barrera Acevedo, Santiago

AU - Dietrich, Heiko

PY - 2019/1

Y1 - 2019/1

N2 - Due to the Hadamard Conjecture, Williamson matrices (WM) and Williamson type matrices (WTM) of order 4n have been primarily investigated for odd n. Several constructions for this case have been introduced, leading to finite and infinite families of WMs and WTMs. The aim of this paper is to present new families of WMs and WTMs with blocks of even order. Let q and r be prime powers congruent to 1 modulo 4. There are WMs of order 4a(q + 1) for every a ∈ {1, 11, 17, 23, 29, 33, 39, 43}. If gcd(q +1, r +1) = 2, then there is a WM of order 2(q +1)(r+1). There are WMs of order 2b(q + 1) and WTMs with circulant blocks of order 2c(q + 1) for every b ∈ {2, . . . , 7} and c ∈ {5, 6}. We prove these results and more by exploiting a recently established correspondence between perfect quaternionic sequences and relative difference sets.

AB - Due to the Hadamard Conjecture, Williamson matrices (WM) and Williamson type matrices (WTM) of order 4n have been primarily investigated for odd n. Several constructions for this case have been introduced, leading to finite and infinite families of WMs and WTMs. The aim of this paper is to present new families of WMs and WTMs with blocks of even order. Let q and r be prime powers congruent to 1 modulo 4. There are WMs of order 4a(q + 1) for every a ∈ {1, 11, 17, 23, 29, 33, 39, 43}. If gcd(q +1, r +1) = 2, then there is a WM of order 2(q +1)(r+1). There are WMs of order 2b(q + 1) and WTMs with circulant blocks of order 2c(q + 1) for every b ∈ {2, . . . , 7} and c ∈ {5, 6}. We prove these results and more by exploiting a recently established correspondence between perfect quaternionic sequences and relative difference sets.

M3 - Article

VL - 73

SP - 207

EP - 219

JO - Australasian Journal of Combinatorics

JF - Australasian Journal of Combinatorics

SN - 1034-4942

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ER -