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The Perfect Binary Sequence of Period 4 for Low Periodic and Aperiodic Autocorrelations

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Sequences, Subsequences, and Consequences

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 4893))

Abstract

The perfect binary sequence of period 4 − ‘0111’ (or cyclic shifts of itself or its complement) − has the optimal periodic autocorrelation function where all out-of-phase values are zero. Not surprisingly, it is also the Barker sequence of length 4 where all out-of-phase aperiodic autocorrelation values have the magnitudes of at most one. From these observations, the applications of the sequence for low periodic and aperiodic autocorrelations are studied. First, the perfect sequence is discussed for binary sequences with optimal periodic autocorrelation. New binary sequences of period N = 4(2m − 1), m = 2k with optimal periodic autocorrelation are presented, which are obtained by a slight modification of product sequences of binary m-sequences and the perfect sequence. Then, it is observed that a product sequence of the Legendre and the perfect sequences has not only the optimal periodic but also the good aperiodic autocorrelations with the asymptotic merit factor 6. Moreover, if the product sequences replace Legendre sequences in Borwein, Choi, Jedwab (BCJ) sequences, or equivalently Kristiansen-Parker sequences (simply BCJ-KP sequences), numerical results show that the resulting sequences have the same asymptotic merit factor as the BCJ-KP sequences.

This work was supported by NSERC Grant RGPIN 227700-00.

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References

  1. Physical Layer Standard for cdma2000 Spread Spectrum Systems, Ver. 1.0, Revision D (February 2004)

    Google Scholar 

  2. 3GPP TS 25.213 v.3.9.0, Technical Specification Group Radio Access Network; Spreading and modulation (FDD) (Release 1999) (December 2003)

    Google Scholar 

  3. IEEE Standard 802.11, Information Technology - Telecommunications and Information Exchange between Systems - Local and Metropolitan Area Networks - Specific Requirements Part 11: Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) Specifications (2003)

    Google Scholar 

  4. Arasu, K.T., Ding, C., Helleseth, T., Kumar, P.V., Martinsen, H.: Almost difference sets and their sequences with optimal autocorrelation. IEEE Trans. Inform. Theory 47(7), 2934–2943 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  5. Barker, R.H.: Group synchronization of binary digital systems. In: Proceedings of the Second London Symposium on Information Theory, pp. 273–287 (1953)

    Google Scholar 

  6. Baumert, L.D.: Cyclic Difference Sets. Springer, Berlin (1971)

    MATH  Google Scholar 

  7. Beenker, G.F.M., Claasen, T.A.C.M., Hermens, P.W.C.: Binary sequences with a maximally flat amplitude spectrum. Philips J. Res 40, 289–304 (1985)

    MATH  Google Scholar 

  8. Borwein, P., Choi, K.-K.S., Jedwab, J.: Binary sequences with merit factor greater than 6.34. IEEE Inform. Theory 50, 3234–3249 (2004)

    Article  MathSciNet  Google Scholar 

  9. Dillon, J.F., Dobbertin, H.: New cyclic difference sets with Singer parameters. Finite Fields and Their Applications 10, 342–389 (2004)

    Google Scholar 

  10. Golay, M.J.E.: Sieves for low autocorrelation binary sequences. IEEE Trans. Inform. Theory IT-23, 43–51 (1977)

    Article  Google Scholar 

  11. Golay, M.J.E.: The merit factor of Legendre sequences. IEEE Trans. Inform. Theory IT-29, 934–936 (1983)

    Article  Google Scholar 

  12. Golomb, S.W., Gong, G.: Signal Design for Good Correlation - for Wireless Communication, Cryptography and Radar. Cambridge University Press, Cambridge (2005)

    MATH  Google Scholar 

  13. Hall, M.: A survey of difference sets. Proc. Amer. Math. Soc. 7, 975–986 (1956)

    Article  MathSciNet  Google Scholar 

  14. Hφholdt, T., Jensen, H.E.: Determination of the merit factor of Legendre sequences. Trans. Inform. Theory 34(1), 161–164 (1988)

    Article  Google Scholar 

  15. Jedwab, J.: A survey of the merit factor problem for binary sequences. In: Helleseth, T., Sarwate, D., Song, H.-Y., Yang, K. (eds.) SETA 2004. LNCS, vol. 3486, pp. 30–55. Springer, Heidelberg (2005)

    Google Scholar 

  16. Jungnickel, D., Pott, A.: Difference sets: An introduction. In: Pott, A., Kumar, P.V., Helleseth, T., Jungnickel, D. (eds.) Difference Sets, Sequences and their Correlation Properties. NATO Science Series C, vol. 542, pp. 259–296 (1999)

    Google Scholar 

  17. Kristiansen, R.: On the Aperiodic Autocorrelation of Binary Sequences, Master’s Thesis, Department of Informatics, University of Bergen (2003)

    Google Scholar 

  18. Kristiansen, R., Parker, M.G.: Binary sequences with merit factor > 6.3. IEEE Trans. Inform. Theory. 50, 3385–3389 (2004)

    Article  MathSciNet  Google Scholar 

  19. Lüke, H.D.: Sequences and arrays with perfect periodic correlation. IEEE Trans. Aerosp. Electron. Syst. 24(3), 287–294 (1988)

    Article  Google Scholar 

  20. No, J.-S., Chung, H., Song, H.-Y., Yang, K., Lee, J.-D., Helleseth, T.: New construction for binary sequences of period p m − 1 with optimal autocorrelation using (z + 1)d + az d + b. IEEE Trans. Inform. Theory 47(4), 1638–1644 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  21. Paley, R.E.A.C.: On orthogonal matrices. J. Math. Phys. 12, 311–320 (1933)

    MATH  Google Scholar 

  22. Parker, M.G.: Even length binary sequence families with low negaperiodic autocorrelation. In: Bozta, S., Sphparlinski, I. (eds.) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. LNCS, vol. 2227, pp. 200–209. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  23. Parker, M.G.: Univariate and multivariate merit factors. In: Helleseth, T., Sarwate, D., Song, H.-Y., Yang, K. (eds.) SETA 2004. LNCS, vol. 3486, pp. 72–100. Springer, Heidelberg (2005)

    Google Scholar 

  24. Parker, M.G., Jedwab, J.: Private Communications (2007)

    Google Scholar 

  25. Paterson, K.G.: Applications of exponential sums in communication theory. In: Walker, M. (ed.) Cryptography and Coding. LNCS, vol. 1746, pp. 1–24. Springer, Heidelberg (1999)

    Chapter  Google Scholar 

  26. Ireland, K., Rosen, M.: A Classical Introduction to Modern Number theory, 2nd edn. Springer, New York (1991)

    Google Scholar 

  27. Schmidt, B.: Cyclotomic integers and finite geometry. J. Amer. Math. Soc. 12, 929–952 (1999)

    MATH  MathSciNet  Google Scholar 

  28. Scholtz, R.A., Pozar, D.M., Namgoong, W.: Ultra-Wideband Radio. EURASIP Journal on Applied Signal Processing 3, 252–272 (2005)

    Article  Google Scholar 

  29. Turyn, R.J., Storer, J.: On binary sequences. Proc. Amer. Math. Soc. 12, 394–399 (1961)

    Article  MATH  MathSciNet  Google Scholar 

  30. Yu., N.Y., Gong, G.: New binary sequences with optimal autocorrelation magnitude. IEEE Trans. Inform. Theory (submitted to) (August 2006) http://www.cacr.math.uwaterloo.ca/techreports/2006/cacr2006-29.pdf

  31. Yu, N.Y., Gong, G.: Applications of the perfect binary sequence of period 4 for low periodic and aperiodic autocorrelations. In: Golomb, S.W., Gong, G., Helleseth, T., Song, H.-Y. (eds.) SEQUENCES 2007. LNCS, vol. 4893, pp. 22–43. Springer, Heidelberg (2007)

    Google Scholar 

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Authors and Affiliations

  1. Department of Electrical and Computer Engineering, University of Waterloo, Waterloo, Ontario, Canada

    Nam Yul Yu & Guang Gong

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  1. Nam Yul Yu
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  2. Guang Gong
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Solomon W. Golomb Guang Gong Tor Helleseth Hong-Yeop Song

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Yu, N.Y., Gong, G. (2007). The Perfect Binary Sequence of Period 4 for Low Periodic and Aperiodic Autocorrelations. In: Golomb, S.W., Gong, G., Helleseth, T., Song, HY. (eds) Sequences, Subsequences, and Consequences. Lecture Notes in Computer Science, vol 4893. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-77404-4_4

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  • DOI: https://doi.org/10.1007/978-3-540-77404-4_4

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-77403-7

  • Online ISBN: 978-3-540-77404-4

  • eBook Packages: Computer ScienceComputer Science (R0)

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