Recent Advances in Low-Correlation Sequences

Chapter

This chapter aims to provide an overview of four topics of current interest relating to the design and analysis of sequences with low correlation. The topics in question are the discovery of new families of cyclic Hadamard difference sets over the past decade following a gap of almost 40 years, the recent realization of the existence of long sequences with larger merit factor than was previously suspected, the development of a theory of sequences possessing a low-correlation zone, and the recent novel construction of low-correlation sequences over the quadrature-amplitude-modulation (QAM) alphabet. While there has also been considerable recent interest on the topic of the design of sequences with low values of peak-to-average power ratio (PAPR), this topic has been addressed in-depth in the recent publication [1] by Litsyn.

Keywords

Radar Autocorrelation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Simon Litsyn, Peak Power Control in Multicarrier Communications, Cambridge University Press, 2007.Google Scholar
  2. 2.
    Solomon W. Golomb and Guang Gong, Signal Design for Good Correlation: For Wireless Communication, Cryptography, and Radar, Cambridge University Press, 2005.Google Scholar
  3. 3.
    D.V. Sarwate and Pursley, M.B., Crosscorrelation properties of pseudorandom and related sequences, Proceedings of the IEEE, May, vol. 68, no. 5, pp. 593-619, 1980.Google Scholar
  4. 4.
    Pingzhi Fan and Mike Darnell, Sequence Design for Communications Applications, Research Studies Press, 1996.Google Scholar
  5. 5.
    B. Gordon, W. H. Mills, and L. R. Welch, “Some new difference sets,” Canad. J. Math, vol. 14, pp. 614–625, 1962.MATHMathSciNetGoogle Scholar
  6. 6.
    R. A. Scholtz and L. R. Welch, “GMW sequences,” IEEE Transactions on Information Theory, vol. 30, no. 3, pp. 548–553, 1984.MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    L. D. Baumert, Cyclic Difference Sets, vol. 182 of Lecture Notes in Mathematics. Berlin-New York: Springer-Verlag, 1971.Google Scholar
  8. 8.
    S. W. Golomb, Shift Register Sequences. Laguna Hills, CA: Aegean Press, 1982.Google Scholar
  9. 9.
    J.-S. No, S. Golomb, G. Gong, H.-K. Lee, and P. Gaal, “Binary pseudorandom sequences of period 2n – 1 with ideal autocorrelation,” IEEE Transactions on Information Theory, vol. 44, pp. 814–817, Mar. 1998.MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    H. Dobbertin, “Kasami power functions, permutation polynomials and cyclic difference sets,” Difference Sets, Sequences and Their Correlation Properties, Eds. A. Pott et. al., pp. 133–158, 1999, Kluwer Academic Publishers.Google Scholar
  11. 11.
    J. Dillon and H. Dobbertin, “New cyclic difference sets with Singer parameters,” Finite Fields and Their Applications, vol. 10, no. 3, pp. 342–389, 2004.MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    J. No, H. Chung, and M. Yun, “Binary pseudorandom sequences of period 2m – 1 with ideal autocorrelation generated by the polynomial \(z^d+(z+1)^d\),” IEEE Transactions on Information Theory, vol. 44, no. 3, pp. 1278–1282, 1998.MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    J. Dillon, “Multiplicative difference sets via additive characters,” Designs, Codes and Cryptography, vol. 17, no. 1, pp. 225–235, 1999.MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    A. Maschietti, “Difference sets and hyperovals,” Designs, Codes and Cryptography, vol. 14, no. 1, pp. 89–98, 1998.MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    R. Turyn and J. Storer, “On binary sequences,” Proc. Amer. Math. Soc, vol. 12, no. 3, pp.394–399, 1961.MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    S. Eliahou and M. Kervaire, “Barker sequences and difference sets,” Énseign. Math., vol. 38, pp. 345–382, 1992.MATHMathSciNetGoogle Scholar
  17. 17.
    J. Jedwab and S. Lloyd, “A note on the nonexistence of Barker sequences,” Designs, Codes and Cryptography, vol. 2, no. 1, pp. 93–97, 1992.MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    M. J. E. Golay, “A class of finite binary sequences with alternate autocorrelation values equal to zero,” IEEE Transactions on Information Theory, vol. 18, no. 3, pp. 449–450, 1972.MATHCrossRefGoogle Scholar
  19. 19.
    D. J. Newman and J. S. Byrnes, “The L 4 norm of a polynomial with coefficients ±1,” Amer. Math. Monthly, vol. 97, pp. 42–45, 1990.CrossRefMathSciNetGoogle Scholar
  20. 20.
    J. M. Jensen, H. E. Jensen, and T. Høholdt, “The merit factor of binary sequences related to difference sets,” IEEE Transactions on Information Theory, vol. 37, no. 3, pp. 617–626, 1991.CrossRefGoogle Scholar
  21. 21.
    S. Mertens and H. Bauke, “Ground States of the Bernasconi model with open boundary conditions,” available online http://odysseus.nat.uni-magdeburg.de/mertens/bernasconi/open.dat, November 2004.
  22. 22.
    J. Knauer, “Merit factor records,” available online http://www.cecm.sfu.ca/jknauer/labs/records.html, Nov. 2004.
  23. 23.
    M. J. E. Golay, “Sieves for low autocorrelation binary sequences,” IEEE Transactions on Information Theory, vol. 23, no. 1, pp. 43–51, 1977.MATHCrossRefGoogle Scholar
  24. 24.
    M. J. E. Golay, “The merit factor of long low autocorrelation binary sequences,” IEEE Transactions on Information Theory, vol. 28, no. 3, pp. 543–549, 1982.CrossRefGoogle Scholar
  25. 25.
    T. Høholdt, H. E. Jensen, and J. Justesen, “Aperiodic correlations and the merit factor of a class of binary sequences,” IEEE Transactions on Information Theory, vol. 31, no. 4, pp. 549–552, 1985.CrossRefGoogle Scholar
  26. 26.
    T. Høholdt and H. E. Jensen, “Determination of the merit factor of Legendre sequences,” IEEE Transactions on Information Theory, vol. 34, no. 1, pp. 161–164, 1988.CrossRefGoogle Scholar
  27. 27.
    P. Borwein and K.-K. S. Choi, “Merit factors of polynomials formed by Jacobi symbols,” Canadian Journal of Mathematics, vol. 53, no. 1, pp. 33–50, 2001.MATHMathSciNetGoogle Scholar
  28. 28.
    M. J. E. Golay, “The merit factor of Legendre sequences,” IEEE Transactions on Information Theory, vol. 29, no. 6, pp. 934–936, 1983.MATHCrossRefGoogle Scholar
  29. 29.
    M. G. Parker, “Even length binary sequence families with low negaperiodic autocorrelation,” Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, AAECC-14 Proceedings, vol. 2227, pp. 200–210, 2001.Google Scholar
  30. 30.
    T. Høholdt, “The merit factor of binary sequences,” Difference Sets, Sequences and Their Correlation Properties, Eds. A. Pott et. al., pp. 227–237, 1999, Kluwer Academic Publishers.Google Scholar
  31. 31.
    P. Borwein, K.-K. S. Choi, and J. Jedwab, “Binary sequences with merit factor greater than 6.34,” IEEE Transactions on Information Theory, vol. 50, no. 12, pp. 3234–3249, 2004.CrossRefMathSciNetGoogle Scholar
  32. 32.
    R. A. Kristiansen and M. G. Parker, “Binary sequences with merit factor > 6.3,” IEEE Transactions on Information Theory, vol. 50, no. 12, pp. 3385–3389, 2004.CrossRefMathSciNetGoogle Scholar
  33. 33.
    S. Boztaş, “CDMA over QAM and other arbitrary energy constellations,” Communication Systems, IEEE International Conference on, vol. 2, pp. 21.7.1–21.7.5, 1996.Google Scholar
  34. 34.
    C. RoBing and V. Tarokh, “A construction of OFDM 16-QAM sequences having low peak powers,” IEEE Transactions on Information Theory, vol. 47, no. 5, pp. 2091–2094, 2001.CrossRefGoogle Scholar
  35. 35.
    H. Lu and P. V. Kumar, “A unified construction of space-time codes with optimal rate-diversity tradeoff,” IEEE Transactions on Information Theory, vol. 51, no. 5, pp. 1709–1730, 2005.CrossRefMathSciNetGoogle Scholar
  36. 36.
    S. Boztaş, R. Hammons, and P. V. Kumar, “4-Phase sequences with near-optimum correlation properties,” IEEE Transactions on Information Theory, vol. 38, no. 3, pp. 1101–1113, 1992.MATHCrossRefGoogle Scholar
  37. 37.
    A. R. Hammons Jr, P. V. Kumar, A. R. Calderbank, N. J. A. Sloane, and P. Sole, “The Z4,–linearity of Kerdock, Preparata, Goethals, and related codes,” IEEE Transactions on Information Theory, vol. 40, no. 2, pp. 301–319, 1994.MATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    T. Helleseth and P. V. Kumar, “Sequences with low correlation,” in Handbook of Coding Theory, Eds. V. Pless and C. Huffman, 1998, Elsevier Science Publishers.Google Scholar
  39. 39.
    P. V. Kumar, T. Helleseth, and A. R. Calderbank, “An upper bound for Weil exponential sums over Galois rings and applications,” IEEE Transactions on Information Theory, vol. 41, no. 2, pp. 456–468, 1995.MATHCrossRefMathSciNetGoogle Scholar
  40. 40.
    P. Sole, “A quaternary cyclic code, and a family of quadriphase sequences with low correlation properties,” Proceedings of the Third International Colloquium on Coding Theory and Applications, pp. 193–201, 1989.Google Scholar
  41. 41.
    K. Yang, T. Helleseth, P. V. Kumar, and A. G. Shanbhag, “On the weight hierarchy of Kerdock codes over Z4,” IEEE Transactions on Information Theory, vol. 42, no. 5, pp. 1587–1593, 1996.MATHCrossRefMathSciNetGoogle Scholar
  42. 42.
    M. Anand and P. V. Kumar, “Low-correlation sequences over the QAM constellation,” IEEE Transactions on Information Theory, vol. 54, no. 2, pp. 791–810, 2008.CrossRefMathSciNetGoogle Scholar
  43. 43.
    G. Garg, P. V. Kumar, and C. E. V. Madhavan, “Low correlation interleaved QAM sequences,” Information Theory, 2008. Proceedings. IEEE International Symposium on, 2008.Google Scholar
  44. 44.
    G. Garg, P. V. Kumar, and C. E. V. Madhavan, “Two new families of low correlation interleaved QAM sequences,” Sequences and Their Applications, International Conference on, 2008.Google Scholar
  45. 45.
    B. Long, P. Zhang, and J. Hu, “A generalized QS-CDMA system and the design of new spreading codes,” IEEE Transactions on Vehicular Technology, vol. 47, no. 4, pp. 1268–1275, 1998.CrossRefGoogle Scholar
  46. 46.
    X. H. Tang, P. Z. Fan, and S. Matsufuji, “Lower bounds on correlation of spreading sequence set with low or zero correlation zone,” Electronics Letters, vol. 36, no. 6, pp. 551–552, 2000.CrossRefGoogle Scholar
  47. 47.
    G. Gong, S. Golomb, and H.-Y. Song, “A note on low correlation zone signal sets,” IEEE Transactions on Information Theory, vol. 53, no. 7, pp. 2575–2581, 2007.CrossRefMathSciNetGoogle Scholar
  48. 48.
    J. Jang, J. No, and H. Chung, “A new construction of optimal p 2-ary low correlation zone sequences using unified sequences,” IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, vol. 89, no. 10, pp. 2656–2661, 2006.CrossRefGoogle Scholar
  49. 49.
    X. H. Tang and P. Z. Fan, “Large families of generalized d-form sequences with low correlations and large linear span based on the interleaved technique,” preprint, 2004.Google Scholar
  50. 50.
    J. Chung and K. Yang, “New design of quaternary low-correlation zone sequence sets and quaternary hadamard matrices,” IEEE Transactions on Information Theory, vol. 54, no. 8, pp. 3733–3737, 2008.CrossRefMathSciNetGoogle Scholar
  51. 51.
    S. Kim, J. Jang, J. No, and H. Chung, “New constructions of quaternary low correlation zone sequences,” IEEE Transactions on Information Theory, vol. 51, no. 4, pp. 1469–1477, 2005.CrossRefMathSciNetGoogle Scholar
  52. 52.
    G. Gong and H.-Y. Song, “Two-tuple-balance of nonbinary sequences with ideal two-level autocorrelation,” Information Theory, 2003. Proceedings. IEEE International Symposium on, p. 404, 29 Jun.–4 Jul. 2003.Google Scholar
  53. 53.
    S.-H. Kim, J.-S. No, H. Chung, and T. Helleseth, “New cyclic relative difference sets constructed from d-homogeneous functions with difference-balanced property,” IEEE Transactions on Information Theory, vol. 51, pp. 1155–1163, March 2005.CrossRefMathSciNetGoogle Scholar
  54. 54.
    G. Gong and H.-Y. Song, “Two-tuple balance of non-binary sequences with ideal two-level autocorrelation,” Discrete Applied Mathematics, vol. 154, no. 18, pp. 2590–2598, 2006.MATHCrossRefMathSciNetGoogle Scholar
  55. 55.
    X. Tang and P. Fan, “A class of pseudonoise sequences over GF (P) with low correlation zone,” IEEE Transactions on Information Theory, vol. 47, no. 4, pp. 1644–1649, 2001.MATHCrossRefMathSciNetGoogle Scholar
  56. 56.
    N. Y. Yu and G. Gong, “The perfect binary sequence of period 4 for low periodic and aperiodic autocorrelation,” Lecture Notes in Computer Science (LNCS), vol. 4893, pp. 37–49, 2007.CrossRefGoogle Scholar
  57. 57.
    J. Jedwab, “A survey of the merit factor problem for binary sequences,” Sequences and their Applications - Proceedings of SETA, vol. 3486, pp. 30–55, 2004.Google Scholar
  58. 58.
    Y. Kim, J. Jang, J. No, and H. Chung, “New design of low-correlation zone sequence sets,” IEEE Transactions on Information Theory, vol. 52, no. 10, pp. 4607–4616, 2006.CrossRefMathSciNetGoogle Scholar
  59. 59.
    X. Tang and P. Udaya, “New construction of low correlation zone sequences from Hadamard matrices,” preprint, 2007.Google Scholar
  60. 60.
    J. Jang, J. No, H. Chung, and X. Tang, “New sets of optimal p-ary low-correlation zone sequences,” IEEE Transactions on Information Theory, vol. 53, no. 2, pp. 815–821, 2007.CrossRefMathSciNetGoogle Scholar
  61. 61.
    J. Chung, J. No, Y. Kim, J. Jang, and H. Chung, “Generalized extending method for construction of q-ary low correlation zone sequence sets,” Information Theory, 2008. Proceedings. IEEE International Symposium on, pp. 1927–1930, 2008.Google Scholar
  62. 62.
    R. De Gaudenzi, C. Elia, and R. Viola, “Bandlimited quasi-synchronous CDMA: A novel satellite access technique for mobile and personal communication systems,” IEEE Journal on Selected Areas in Communications, vol. 10, no. 2, pp. 328–343, 1992.CrossRefGoogle Scholar
  63. 63.
    J. Jang, J. Chung, and J. No, “Quaternary low correlation zone sequence set with flexible parameters,” Information Theory, 2008. Proceedings. IEEE International Symposium on, pp. 2767–2771, 2008.Google Scholar
  64. 64.
    J. Yang, X. Jin, K. Song, J. No, and D. Shin, “Multicode MIMO systems with quaternary LCZ and ZCZ sequences,” IEEE Transactions on Vehicular Technology, vol. 57, no. 4, pp. 2334–2341, 2008.CrossRefGoogle Scholar
  65. 65.
    H. Torii, M. Nakamura, and N. Suehiro, “A new class of polyphase sequence sets with optimal zero-correlation zones,” IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, no. 7, pp. 1987–1994, 2005.Google Scholar
  66. 66.
    T. Hayashi and S. Matsufuji, “On optimal construction of two classes of ZCZ codes,” IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, vol. 89, no. 9, pp. 2345–2350, 2006.CrossRefGoogle Scholar
  67. 67.
    T. Hayashi, “Zero-correlation zone sequence set construction using an even-perfect sequence and an odd-perfect sequence,” IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, vol. 90, no. 9, pp. 1871–1875, 2007.CrossRefGoogle Scholar
  68. 68.
    T. Hayashi, “A novel class of zero-correlation zone sequence sets constructed from a perfect sequence,” IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, vol. 91, no. 4, pp. 1233–1237, 2008.CrossRefGoogle Scholar
  69. 69.
    Z. Zhou, X. Tang, and G. Gong, “A new class of sequences with zero or low correlation zone based on interleaving technique,” IEEE Transactions on Information Theory, vol. 54, no. 9, pp. 4267–4273, 2008.CrossRefMathSciNetGoogle Scholar
  70. 70.
    X. Tang and W. H. Mow, “A new systematic construction of zero correlation zone sequences based on interleaved perfect sequences,” preprint, 2008.Google Scholar
  71. 71.
    F. MacWilliams and N. Sloane, “Pseudo-random sequences and arrays,” Proceedings of the IEEE, vol. 64, pp. 1715–1729, Dec. 1976.CrossRefMathSciNetGoogle Scholar
  72. 72.
    M. Antweiler, L. Bomer, and H.-D. Luke, “Perfect ternary arrays,” IEEE Transactions on Information Theory, vol. 36, pp. 696–705, May 1990.MATHCrossRefGoogle Scholar
  73. 73.
    P. V. Kumar, R. A. Scholtz, and L. R. Welch, “Generalized bent functions and their properties,” Journal of Combinatorial Theory. Series A, vol. 40, pp. 90–107, 1985.MATHCrossRefMathSciNetGoogle Scholar
  74. 74.
    N. Suehiro, “A signal design without co-channel interference for approximately synchronized CDMA systems,” IEEE Journal on Selected Areas in Communications, vol. 12, no. 5, pp. 837–841, 1994.CrossRefGoogle Scholar
  75. 75.
    P. Z. Fan, N. Suehiro, N. Kuroyanagi, and X. M. Deng, “Class of binary sequences with zero correlation zone,” Electronics Letters, vol. 35, no. 10, pp. 777–779, 1999.CrossRefGoogle Scholar
  76. 76.
    H. Torii, M. Nakamura, and N. Suehiro, “A new class of zero-correlation zone sequences,” IEEE Transactions on Information Theory, vol. 50, pp. 559–565, Mar. 2004.CrossRefMathSciNetGoogle Scholar
  77. 77.
    H. Torii and M. Nakamura, “Enhancement of ZCZ sequence set construction procedure,” IEICE Transactions on Fundamentals of Electronics, Communications and Computer Science, vol. 90, no. 2, pp. 535–538, 2007.CrossRefGoogle Scholar
  78. 78.
    D. Peng, P. Fan, and N. Suehiro, “Construction of sequences with large zero correlation zone,” IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, vol. 88, no. 11, pp. 3256–3259, 2005.CrossRefGoogle Scholar
  79. 79.
    X. Tong and Q. Wen, “New constructions of zcz sequence set with large family size,” Signal Design and Its Applications in Communications, 2007. IWSDA 2007. 3rd International Workshop on, pp. 99–103, Sept. 2007.Google Scholar
  80. 80.
    T. Hayashi, “Binary zero-correlation zone sequence set construction using a primitive linear recursion,” IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, no. 7, pp. 2034–2038, 2005.Google Scholar
  81. 81.
    T. Hayashi, “Ternary sequence set having periodic and aperiodic zero-correlation zone,” IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, vol. 89, no. 6, pp. 1825–1831, 2006.CrossRefGoogle Scholar
  82. 82.
    T. Hayashi, “Binary zero-correlation zone sequence set construction using a cyclic hadamard sequence,” IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, vol. 89, no. 10, pp. 2649–2655, 2006.CrossRefGoogle Scholar
  83. 83.
    T. Hayashi, “Binary zero-correlation zone sequence set constructed from an M-sequence,” IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, no. 2, pp. 633–638, 2006.Google Scholar
  84. 84.
    T. Hayashi, “An integrated sequence construction of binary zero-correlation zone sequences,” IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, vol. 90, no. 10, pp. 2329–2335, 2007.CrossRefGoogle Scholar
  85. 85.
    T. Hayashi, “Zero-correlation zone sequence set constructed from a perfect sequence,” IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, vol. 90, no. 5, pp. 1107–1111, 2007.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag US 2009

Authors and Affiliations

  1. 1.Department of Computer Science and AutomationIndian Institute of ScienceBangaloreIndia
  2. 2.Department of InformaticsUniversity of BergenBergenNorway
  3. 3.Department of Electrical Communication EngineeringIndian Institute of ScienceBangaloreIndia

Personalised recommendations