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Designs, Codes and Cryptography

, Volume 74, Issue 2, pp 365–377 | Cite as

Compression of periodic complementary sequences and applications

  • Dragomir Ž. Ɖoković
  • Ilias S. Kotsireas
Article

Abstract

A collection of complex sequences of length \(v\) is complementary if the sum of their periodic autocorrelation function values at all non-zero shifts is constant. For a complex sequence \(A = [a_0,a_1,\ldots ,a_{v-1}]\) of length \(v = dm\) we define the \(m\)-compressed sequence \(A^{(d)}\) of length \(d\) whose terms are the sums \(a_i + a_{i+d} + \cdots + a_{i+(m-1)d}\). We prove that the \(m\)-compression of a complementary collection of sequences is also complementary. The compression procedure can be used to simplify the construction of complementary \(\{\pm 1\}\)-sequences of composite length. In particular, we construct several supplementary difference sets \((v;r,s;\lambda )\) with \(v\) even and \(\lambda = (r+s)-v/2\), given here for the first time. There are 15 normalized parameter sets \((v;r,s;\lambda )\) with \(v\le 50\) for which the existence question was open. We resolve all but one of these cases.

Keywords

Supplementary difference sets Binary sequences Periodic autocorrelation Power spectral density  

Mathematics Subject Classification

05B20 05B30 

Notes

Acknowledgments

The authors thank Joe Sawada and Daniel Recoskie for sharing improved versions of their C code for computing ordinary and charmed bracelets. We also thank William Orrick for providing reference [18]. The authors wish to acknowledge generous support by NSERC. This work was made possible by the facilities of the Shared Hierarchical Academic Research Computing Network (SHARCNET) and Compute/Calcul Canada.

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Pure MathematicsUniversity of WaterlooWaterlooCanada
  2. 2.Department of Physics & Computer ScienceWilfrid Laurier UniversityWaterlooCanada

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