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Recent Advances in Low-Correlation Sequences

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New Directions in Wireless Communications Research

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.

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References

  1. Simon Litsyn, Peak Power Control in Multicarrier Communications, Cambridge University Press, 2007.

    Google Scholar 

  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. 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. Pingzhi Fan and Mike Darnell, Sequence Design for Communications Applications, Research Studies Press, 1996.

    Google Scholar 

  5. B. Gordon, W. H. Mills, and L. R. Welch, “Some new difference sets,” Canad. J. Math, vol. 14, pp. 614–625, 1962.

    MATH  MathSciNet  Google Scholar 

  6. R. A. Scholtz and L. R. Welch, “GMW sequences,” IEEE Transactions on Information Theory, vol. 30, no. 3, pp. 548–553, 1984.

    Article  MATH  MathSciNet  Google Scholar 

  7. L. D. Baumert, Cyclic Difference Sets, vol. 182 of Lecture Notes in Mathematics. Berlin-New York: Springer-Verlag, 1971.

    Google Scholar 

  8. S. W. Golomb, Shift Register Sequences. Laguna Hills, CA: Aegean Press, 1982.

    Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  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. 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.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  13. J. Dillon, “Multiplicative difference sets via additive characters,” Designs, Codes and Cryptography, vol. 17, no. 1, pp. 225–235, 1999.

    Article  MATH  MathSciNet  Google Scholar 

  14. A. Maschietti, “Difference sets and hyperovals,” Designs, Codes and Cryptography, vol. 14, no. 1, pp. 89–98, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  15. R. Turyn and J. Storer, “On binary sequences,” Proc. Amer. Math. Soc, vol. 12, no. 3, pp.394–399, 1961.

    Article  MATH  MathSciNet  Google Scholar 

  16. S. Eliahou and M. Kervaire, “Barker sequences and difference sets,” Énseign. Math., vol. 38, pp. 345–382, 1992.

    MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  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.

    Article  Google Scholar 

  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. J. Knauer, “Merit factor records,” available online http://www.cecm.sfu.ca/jknauer/labs/records.html, Nov. 2004.

  23. M. J. E. Golay, “Sieves for low autocorrelation binary sequences,” IEEE Transactions on Information Theory, vol. 23, no. 1, pp. 43–51, 1977.

    Article  MATH  Google Scholar 

  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.

    Article  Google Scholar 

  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.

    Article  Google Scholar 

  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.

    Article  Google Scholar 

  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.

    MATH  MathSciNet  Google Scholar 

  28. M. J. E. Golay, “The merit factor of Legendre sequences,” IEEE Transactions on Information Theory, vol. 29, no. 6, pp. 934–936, 1983.

    Article  MATH  Google Scholar 

  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. 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. 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.

    Article  MathSciNet  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  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. 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.

    Article  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  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.

    Article  MATH  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  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. 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.

    Article  MATH  MathSciNet  Google Scholar 

  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. 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.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  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. 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. 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.

    Article  Google Scholar 

  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.

    Article  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  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.

    Article  Google Scholar 

  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. 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.

    Article  MathSciNet  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  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. 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.

    Article  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  Google Scholar 

  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. 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.

    Article  MathSciNet  Google Scholar 

  59. X. Tang and P. Udaya, “New construction of low correlation zone sequences from Hadamard matrices,” preprint, 2007.

    Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  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. 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.

    Article  Google Scholar 

  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. 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.

    Article  Google Scholar 

  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. 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.

    Article  Google Scholar 

  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.

    Article  Google Scholar 

  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.

    Article  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  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. F. MacWilliams and N. Sloane, “Pseudo-random sequences and arrays,” Proceedings of the IEEE, vol. 64, pp. 1715–1729, Dec. 1976.

    Article  MathSciNet  Google Scholar 

  72. M. Antweiler, L. Bomer, and H.-D. Luke, “Perfect ternary arrays,” IEEE Transactions on Information Theory, vol. 36, pp. 696–705, May 1990.

    Article  MATH  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  Google Scholar 

  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.

    Article  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  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.

    Article  Google Scholar 

  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.

    Article  Google Scholar 

  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. 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. 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.

    Article  Google Scholar 

  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.

    Article  Google Scholar 

  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. 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.

    Article  Google Scholar 

  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.

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Computer Science and Automation, Indian Institute of Science, Bangalore, 560012, India

    Gagan Garg

  2. Department of Informatics, University of Bergen, HIB, N-5020, Bergen, Norway

    Tor Helleseth

  3. Department of Electrical Communication Engineering, Indian Institute of Science, Bangalore, 560012, India

    P. Vijay Kumar

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  1. Gagan Garg
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  2. Tor Helleseth
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  3. P. Vijay Kumar
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Correspondence to Gagan Garg .

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Editors and Affiliations

  1. School of Engineering &, Harvard University, Oxford St. 33, Cambridge, 02138, U.S.A.

    Vahid Tarokh

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Garg, G., Helleseth, T., Kumar, P.V. (2009). Recent Advances in Low-Correlation Sequences. In: Tarokh, V. (eds) New Directions in Wireless Communications Research. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-0673-1_3

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