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.
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© 2012 Springer-Verlag Berlin Heidelberg
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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. https://doi.org/10.1007/978-3-642-30615-0_15
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DOI: https://doi.org/10.1007/978-3-642-30615-0_15
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-30614-3
Online ISBN: 978-3-642-30615-0
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