Inflation of perfect arrays over the basic quaternions of size mn = (q+1)/2

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Abstract

Arasu and de Launey showed that every perfect quaternary array (that is perfect arrays over the four roots of unity ±1,±i±1,±i ) of size m×nm×n , can be inflated into another perfect quaternary array of size mp×npmp×np , provided p=mn−1p=mn−1 is s prime number. Likewise, they showed that every perfect quaternary array of size m×nm×n , can be inflated into another perfect quaternary array of size mq×nqmq×nq , provided q=2mn−1q=2mn−1 is a prime number and q≡3(mod 4)q≡3(mod 4) . Following from Arasu and de Launey’s first construction, Barrera Acevedo and Jolly showed that every perfect array over the basic quaternions, {1,−1,i,−i,j,−j,k,−k}{1,−1,i,−i,j,−j,k,−k} , of sizes m×nm×n , can be inflated into a new perfect array over the basic quaternions of size mp×npmp×np , provided p=mn−1p=mn−1 is s prime number. Combining this construction with the existence of infinitely many modified Lee sequences over {1,−1,i,−i,j}{1,−1,i,−i,j} (in the sense of Barrera Acevedo and Hall), they showed the existence of infinitely many perfect arrays over the basic quaternions, with appearances of all the basic quaternion elements 1,−1,i,−i,j,−j,k1,−1,i,−i,j,−j,k and −k−k . In this work, we show that every perfect array over the basic quaternions, of size m×nm×n , can be inflated into a perfect quaternary array of size mq×nqmq×nq , provided q=2mn−1q=2mn−1 is a prime number and q≡3(mod 4)q≡3(mod 4)
Original languageEnglish
Title of host publicationSequences and Their Applications - SETA 2014
EditorsKai-Uwe Schmidt, Arne Winterhof
Place of PublicationCham Switzerland
PublisherSpringer
Pages123-133
Number of pages11
Volume8865
ISBN (Electronic)9783319123240
ISBN (Print)9783319123240
DOIs
Publication statusPublished - 2014
EventInternational Conference on Sequences and their Applications 2014 - Melbourne, Australia
Duration: 24 Nov 201428 Nov 2014
Conference number: 8th
https://link.springer.com/book/10.1007/978-3-319-12325-7

Publication series

NameLecture Notes in Computer Science
PublisherSpringer
Volume8865
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

ConferenceInternational Conference on Sequences and their Applications 2014
Abbreviated titleSETA 2014
CountryAustralia
CityMelbourne
Period24/11/1428/11/14
Internet address

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