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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Kuznetsov, O.: Perfect sequences over the real quaternions. In: Fourth International Workshop on Signal Design and its Applications in Communications, IWSDA 2009, vol. 1, pp. 17–20 (2010)
Kuznetsov, O., Hall, T.: Perfect sequences over the real quaternions of longer length. The Online Journal on Mathematics and Statistics 1, 8–11 (2009); The 2010 World Congress on Mathematics and Statistics, WCMS 2010
Lee, C.E.: On a new class of 5-ary sequences exhibiting ideal periodic autocorrelation properties with applications to spread spectrum systems. PhD dissertation. Mississippi State University (1986)
Lee, C.E.: Perfect q-ary sequences from multiplicative characters over GF(p). Electronic Letters 28, 833–835 (1992)
Luke, H.D.: BTP transform and perfect sequences with small phase alphabet. IEEE Trans. Aerosp. Electro. Syst. 32, 497–499 (1996)
Luke, H.D.: Binary and quadriphase sequences with optimal autocorrelation properties: A survey. IEEE Transactions on Information Theory 49, 3271–3282 (2003)
Mow, W.H.: A Unified Construction of Perfect Polyphase Sequences. In: IEEE Proceedings of International Symposium on Information Theory, vol. 1, pp. 495–495 (1995)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
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
Download citation
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
eBook Packages: Computer ScienceComputer Science (R0)