Perfect Sequences of Unbounded Lengths over the Basic Quaternions

  • Santiago Barrera Acevedo
  • Thomas E. Hall
Conference paper

DOI: 10.1007/978-3-642-30615-0_15

Part of the Lecture Notes in Computer Science book series (LNCS, volume 7280)
Cite this paper as:
Acevedo S.B., Hall T.E. (2012) Perfect Sequences of Unbounded Lengths over the Basic Quaternions. In: Helleseth T., Jedwab J. (eds) Sequences and Their Applications – SETA 2012. SETA 2012. Lecture Notes in Computer Science, vol 7280. Springer, Berlin, Heidelberg

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.

Keywords

Perfect Sequences over the Basic Quaternions Lee Sequences Perfect Autocorrelation Perfect Sequences Quaternions 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Santiago Barrera Acevedo
    • 1
  • Thomas E. Hall
    • 1
  1. 1.Monash UniversityAustralia

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