Perfect sequences of unbounded lengths over the basic quaternions

Santiago Barrera Acevedo, Thomas Eric Hall

Research output: Chapter in Book/Report/Conference proceedingConference PaperResearchpeer-review

5 Citations (Scopus)

Abstract

In this paper we show the existence of perfect sequences, of unbounded lengths, over the basic quaternions 1, -1,i, -i,j, -j,k, -k . Perfect sequences over the quaternion algebra were first introduced in 2009. One year later, a perfect sequence of length 5,354,228,880, over a quaternion alphabet with 24 elements, was shown. At this point two main questions were stated: Are there perfect sequences of unbounded lengths over the quaternion algebra? If so, is it possible to restrict the alphabet size to a small one? We answer these two questions by proving that any Lee sequence can always be converted into a sequence over the basic quaternions, which is an alphabet with 8 elements, and then by using the existence of Lee sequences of unbounded lengths to prove the existence of perfect sequences of unbounded lengths over the basic quaternions.
Original languageEnglish
Title of host publicationSequences and Their Applications - SETA 2012
EditorsTor Helleseth, Jonathan Jedwab
Place of PublicationNew York USA
PublisherSpringer
Pages159 - 167
Number of pages9
ISBN (Print)9783642306143
DOIs
Publication statusPublished - 2012
EventInternational Conference on Sequences and their Applications, 2012 - Waterloo, Canada
Duration: 1 Jan 2012 → …

Conference

ConferenceInternational Conference on Sequences and their Applications, 2012
CountryCanada
CityWaterloo
Period1/01/12 → …

Cite this

Barrera Acevedo, S., & Hall, T. E. (2012). Perfect sequences of unbounded lengths over the basic quaternions. In T. Helleseth, & J. Jedwab (Eds.), Sequences and Their Applications - SETA 2012 (pp. 159 - 167). New York USA: Springer. https://doi.org/10.1007/978-3-642-30615-0_15
Barrera Acevedo, Santiago ; Hall, Thomas Eric. / Perfect sequences of unbounded lengths over the basic quaternions. Sequences and Their Applications - SETA 2012. editor / Tor Helleseth ; Jonathan Jedwab. New York USA : Springer, 2012. pp. 159 - 167
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title = "Perfect sequences of unbounded lengths over the basic quaternions",
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Barrera Acevedo, S & Hall, TE 2012, Perfect sequences of unbounded lengths over the basic quaternions. in T Helleseth & J Jedwab (eds), Sequences and Their Applications - SETA 2012. Springer, New York USA, pp. 159 - 167, International Conference on Sequences and their Applications, 2012, Waterloo, Canada, 1/01/12. https://doi.org/10.1007/978-3-642-30615-0_15

Perfect sequences of unbounded lengths over the basic quaternions. / Barrera Acevedo, Santiago; Hall, Thomas Eric.

Sequences and Their Applications - SETA 2012. ed. / Tor Helleseth; Jonathan Jedwab. New York USA : Springer, 2012. p. 159 - 167.

Research output: Chapter in Book/Report/Conference proceedingConference PaperResearchpeer-review

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AB - In this paper we show the existence of perfect sequences, of unbounded lengths, over the basic quaternions 1, -1,i, -i,j, -j,k, -k . Perfect sequences over the quaternion algebra were first introduced in 2009. One year later, a perfect sequence of length 5,354,228,880, over a quaternion alphabet with 24 elements, was shown. At this point two main questions were stated: Are there perfect sequences of unbounded lengths over the quaternion algebra? If so, is it possible to restrict the alphabet size to a small one? We answer these two questions by proving that any Lee sequence can always be converted into a sequence over the basic quaternions, which is an alphabet with 8 elements, and then by using the existence of Lee sequences of unbounded lengths to prove the existence of perfect sequences of unbounded lengths over the basic quaternions.

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Barrera Acevedo S, Hall TE. Perfect sequences of unbounded lengths over the basic quaternions. In Helleseth T, Jedwab J, editors, Sequences and Their Applications - SETA 2012. New York USA: Springer. 2012. p. 159 - 167 https://doi.org/10.1007/978-3-642-30615-0_15