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Designs, Codes and Cryptography

, Volume 78, Issue 1, pp 237–267 | Cite as

Sequences with small correlation

  • Kai-Uwe SchmidtEmail author
Article

Abstract

The extent to which a sequence of finite length differs from a shifted version of itself is measured by its aperiodic autocorrelations. Of particular interest are sequences whose entries are 1 or \(-1\), called binary sequences, and sequences whose entries are complex numbers of unit magnitude, called unimodular sequences. Since the 1950s, there is sustained interest in sequences with small aperiodic autocorrelations relative to the sequence length. One of the main motivations is that a sequence with small aperiodic autocorrelations is intrinsically suited for the separation of signals from noise, and therefore has natural applications in digital communications. This survey reviews the state of knowledge concerning the two central problems in this area: How small can the aperiodic autocorrelations of a binary or a unimodular sequence collectively be and how can we efficiently find the best such sequences? Since the analysis and construction of sequences with small aperiodic autocorrelations is closely tied to the (often much easier) analysis of periodic autocorrelation properties, several fundamental results on corresponding problems in the periodic setting are also reviewed.

Keywords

Aperiodic Autocorrelation Barker Difference set Golay Sequence 

Mathematics Subject Classification

94A55 11B83 05B10 

Notes

Acknowledgments

I would like to thank Christian Günther, Jonathan Jedwab, Dieter Jungnickel, and Peter Wild for some careful comments on a draft of this survey.

References

  1. 1.
    Alon N., Litsyn S., Shpunt A.: Typical peak sidelobe level of binary sequences. IEEE Trans. Inf. Theory 56(1), 545–554 (2010).Google Scholar
  2. 2.
    Antweiler M., Bömer L.: Merit factor of Chu and Frank sequences. IEE Electron. Lett. 46(25), 2068–2070 (1990).Google Scholar
  3. 3.
    Arasu K.T., Ding C., Helleseth T., Kumar P.V., Martinsen H.M.: Almost difference sets and their sequences with optimal autocorrelation. IEEE Trans. Inf. Theory 47(7), 2934–2943 (2001).Google Scholar
  4. 4.
    Baden J.M.: Efficient optimization of the merit factor of long binary sequences. IEEE Trans. Inf. Theory 57(12), 8084–8094 (2011).Google Scholar
  5. 5.
    Barker R.H.: Group synchronization of binary digital systems. In: Jackson W. (ed.) Communication Theory, pp. 173–187. Academic Press, New York (1953).Google Scholar
  6. 6.
    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(5), 289–304 (1985).Google Scholar
  7. 7.
    Bernasconi J.: Low autocorrelation binary sequences: statistical mechanics and configuration state analysis. J. Phys. 48(4), 559–567 (1987).Google Scholar
  8. 8.
    Boehmer A.M.: Binary pulse compression codes. IEEE Trans. Inf. Theory IT 13(2), 156–167 (1967).Google Scholar
  9. 9.
    Borwein P.: Computational Excursions in Analysis and Number Theory. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, vol. 10. Springer-Verlag, New York (2002).Google Scholar
  10. 10.
    Borwein P., Choi K.-K.S.: Merit factors of character polynomials. J. Lond. Math. Soc. 61, 706–720 (2000).Google Scholar
  11. 11.
    Borwein P.B., Ferguson R.A.: A complete description of Golay pairs for lengths up to 100. Math. Comput. 73(246), 967–985 (2004).Google Scholar
  12. 12.
    Borwein P., Mossinghoff M.: Rudin–Shapiro-like polynomials in \(L_4\). Math. Comput. 69(231), 1157–1166 (2000).Google Scholar
  13. 13.
    Borwein P., Mossinghoff M.J.: Wieferich pairs and Barker sequences, II. LMS J. Comput. Math. 17(1), 24–32 (2014).Google Scholar
  14. 14.
    Brauer A.: On a new class of Hadamard determinants. Math. Z. 58, 219–225 (1953).Google Scholar
  15. 15.
    Broughton W.J.: A note on Table I of: “Barker sequences and difference sets”. Enseign. Math. 2, 40(1–2), 105–107 (1994).Google Scholar
  16. 16.
    Cai Y., Ding C.: Binary sequences with optimal autocorrelation. Theor. Comput. Sci. 410(24–25), 2316–2322 (2009).Google Scholar
  17. 17.
    Chu D.: Polyphase codes with good periodic correlation properties. IEEE Trans. Inf. Theory IT-18(4), 531–532 (1972).Google Scholar
  18. 18.
    Davis J.A., Jedwab J.: Peak-to-mean power control in OFDM, Golay complementary sequences, and Reed-Muller codes. IEEE Trans. Inf. Theory 45(7), 2397–2417 (1999).Google Scholar
  19. 19.
    Dillon J.F., Dobbertin H.: New cyclic difference sets with Singer parameters. Finite Fields Appl. 10(3), 342–389 (2004).Google Scholar
  20. 20.
    Ding C., Helleseth T., Lam K.Y.: Several classes of binary sequences with three-level autocorrelation. IEEE Trans. Inf. Theory 45(7), 2606–2612 (1999).Google Scholar
  21. 21.
    Dmitriev D., Jedwab J.: Bounds on the growth rate of the peak sidelobe level of binary sequences. Adv. Math. Commun. 1(4), 461–475 (2007).Google Scholar
  22. 22.
    Ein-Dor L., Kanter I., Kinzel W.: Low autocorrelated multiphase sequences. Phys. Rev. E, 65(2), 020102.1–020102.4 (2002).Google Scholar
  23. 23.
    Eliahou Sh., Kervaire M.: Barker sequences and difference sets. Enseign. Math. (2), 38(3–4), 345–382 (1992).Google Scholar
  24. 24.
    Eliahou Sh., Kervaire M., Saffari B.: A new restriction on the lengths of Golay complementary sequences. J. Comb. Theory Ser. A 55(1), 49–59 (1990).Google Scholar
  25. 25.
    Eliahou Sh., Kervaire M., Saffari B.: On Golay polynomial pairs. Adv. Appl. Math. 12(3), 235–292 (1991).Google Scholar
  26. 26.
    Erdős, P.: Some old and new problems in approximation theory: research problems 95–1. Constr. Approx. 11(3), 419–421 (1995)Google Scholar
  27. 27.
    Evans R., Hollmann H.D.L., Krattenthaler Ch., Xiang Q.: Gauss sums, Jacobi sums, and \(p\)-ranks of cyclic difference sets. J. Comb. Theory Ser. A 87(1), 74–119 (1999).Google Scholar
  28. 28.
    Fiedler F.: Small Golay sequences. Adv. Math. Commun. 7(4), 379–407 (2013).Google Scholar
  29. 29.
    Fiedler F., Jedwab J., Parker M.G.: A framework for the construction of Golay sequences. IEEE Trans. Inf. Theory 54(7), 3114–3129 (2008).Google Scholar
  30. 30.
    Fiedler F., Jedwab J., Parker M.G.: A multi-dimensional approach to the construction and enumeration of Golay complementary sequences. J. Comb. Theory Ser. A 115(5), 753–776 (2008).Google Scholar
  31. 31.
    Fiedler F., Jedwab J., Wiebe A.: A new source of seed pairs for Golay sequences of length \(2^m\). J. Comb. Theory Ser. A 117(5), 589–597 (2010).Google Scholar
  32. 32.
    Frank R.L.: Polyphase complementary codes. IEEE Trans. Inf. Theory 26(6), 641–647 (1980).Google Scholar
  33. 33.
    Frank R., Zadoff S.: Phase shift pulse codes with good periodic correlation properties. IRE Trans. Inf. Theory IT-8(6), 381–382 (1962).Google Scholar
  34. 34.
    Fredman M.L., Saffari B., Smith B.: Polynômes réciproques: conjecture d’Erdős en norme \(L^4,\) taille des autocorrélations et inexistence des codes de Barker. C. R. Acad. Sci. Paris Ser. I Math. 308(15), 461–464 (1989).Google Scholar
  35. 35.
    Gibson R.G., Jedwab J.: Quaternary Golay sequence pairs II: odd length. Des. Codes Cryptogr. 59(1–3), 147–157 (2011).Google Scholar
  36. 36.
    Golay M.J.E.: Static multislit spectrometry and its applications to the panoramic display of infrared spectra. J. Opt. Soc. Am. 41, 468–472 (1951).Google Scholar
  37. 37.
    Golay M.J.E.: Complementary series. IRE Trans. Inf. Theory IT-7(2), 82–87 (1961).Google Scholar
  38. 38.
    Golay M.J.E.: A class of finite binary sequences with alternate autocorrelation values equal to zero. IEEE Trans. Inf. Theory IT-18(3), 449–450 (1972).Google Scholar
  39. 39.
    Golay M.J.E.: The merit factor of long low autocorrelation binary sequences. IEEE Trans. Inf. Theory 28(3), 543–549 (1982).Google Scholar
  40. 40.
    Golomb S.W.: Shift register sequences. Holden-Day Inc., San Francisco (1967).Google Scholar
  41. 41.
    Golomb S.W., Gong G.: Signal Design for Good Correlation. For Wireless Communication, Cryptography, and Radar. Cambridge University Press, Cambridge (2005).Google Scholar
  42. 42.
    Golomb S.W., Scholtz R.A.: Generalized Barker sequences. IEEE Trans. Inf. Theory, IT-11(4), 533–537 (1965).Google Scholar
  43. 43.
    Gordon B., Mills W.H., Welch L.R.: Some new difference sets. Can. J. Math. 14, 614–625 (1962).Google Scholar
  44. 44.
    Günther Ch., Schmidt K.-U.: Merit factors of polynomials derived from difference sets. arXiv:1503.05858 [math.CO].
  45. 45.
    Hall Jr. M.: A survey of difference sets. Proc. Am. Math. Soc. 7, 975–986 (1956).Google Scholar
  46. 46.
    Heimiller R.C.: Phase shift pulse codes with good periodic correlation properties. IRE Trans. Inf. Theory IT-7(4), 254–257 (1961).Google Scholar
  47. 47.
    Helleseth T., Kumar P.V.: Sequences with low correlation. In: Handbook of Coding Theory, vol. II, pp. 1765–1853. North-Holland, Amsterdam (1998).Google Scholar
  48. 48.
    Høholdt T.: The merit factor problem for binary sequences. In: Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. Lecture Notes in Computer Science, vol. 3857, pp. 51–59. Springer, Berlin (2006).Google Scholar
  49. 49.
    Høholdt T., Jensen H.E.: Determination of the merit factor of Legendre sequences. IEEE Trans. Inf. Theory 34(1), 161–164 (1988).Google Scholar
  50. 50.
    Høholdt T., Jensen H.E., Justesen J.: Aperiodic correlations and the merit factor of a class of binary sequences. IEEE Trans. Inf. Theory 31(4), 549–552 (1985).Google Scholar
  51. 51.
    Holzmann W.H., Kharaghani H.: A computer search for complex Golay sequences. Australas. J. Comb. 10, 251–258 (1994).Google Scholar
  52. 52.
    Jedwab J.: A survey of the merit factor problem for binary sequences. In: Proceedings of Sequences and Their Applications. Lecture Notes in Computer Science, vol. 3486, pp. 30–55. Springer Verlag, New York (2005).Google Scholar
  53. 53.
    Jedwab J.: What can be used instead of a Barker sequence? Contemp. Math. 461, 153–178 (2008).Google Scholar
  54. 54.
    Jedwab J., Katz D.J., Schmidt K.-U.: Advances in the merit factor problem for binary sequences. J. Comb. Theory Ser. A 120(4), 882–906 (2013).Google Scholar
  55. 55.
    Jedwab J., Katz D.J., Schmidt K.-U.: Littlewood polynomials with small \(L^4\) norm. Adv. Math. 241, 127–136 (2013).Google Scholar
  56. 56.
    Jedwab J., Parker M.G.: A construction of binary Golay sequence pairs from odd-length Barker sequences. J. Comb. Des. 17(6), 478–491 (2009).Google Scholar
  57. 57.
    Jedwab J., Yoshida K.: The peak sidelobe level of families of binary sequences. IEEE Trans. Inf. Theory 52(5), 2247–2254 (2006).Google Scholar
  58. 58.
    Jensen J.M., Jensen H.E., Høholdt T.: The merit factor of binary sequences related to difference sets. IEEE Trans. Inf. Theory 37(3), 617–626 (1991).Google Scholar
  59. 59.
    Jungnickel D., Pott A.: Perfect and almost perfect sequences. Discret. Appl. Math. 95(1–3), 331–359 (1999).Google Scholar
  60. 60.
    Kumar P.V., Scholtz R.A., Welch L.R.: Generalized bent functions and their properties. J. Comb. Theory Ser. A 40(1), 90–107 (1985).Google Scholar
  61. 61.
    Lander E.S.: Symmetric designs: an algebraic approach. London Mathematical Society Lecture Note Series, vol. 74. Cambridge University Press, Cambridge (1983).Google Scholar
  62. 62.
    Lempel A., Cohn M., Eastman W.L.: A class of balanced binary sequences with optimal autocorrelation properties. IEEE Trans. Inf. Theory IT-23(1), 38–42 (1977).Google Scholar
  63. 63.
    Leukhin A.N., Potekhin E.N.: Exhaustive search for optimal minimum peak sidelobe binary sequences up to length 80. In: Sequences and Their Applications. Lecture Notes in Computer Science, vol. 8865, pp. 157–169. Springer, New York (2014).Google Scholar
  64. 64.
    Leung K.H., Schmidt B.: The field descent method. Des. Codes Cryptogr. 36(2), 171–188 (2005).Google Scholar
  65. 65.
    Leung K.H., Schmidt B.: New restrictions on possible orders of circulant Hadamard matrices. Des. Codes Cryptogr. 64(1–2), 143–151 (2012).Google Scholar
  66. 66.
    Leung K.H., Schmidt B.: The anti-field-descent method. J. Comb. Theory Ser. A (to appear)Google Scholar
  67. 67.
    Littlewood J.E.: On the mean values of certain trigonometric polynomials. J. Lond. Math. Soc. 36, 307–334 (1961).Google Scholar
  68. 68.
    Littlewood J.E.: On the mean values of certain trigonometric polynomials II. Ill. J. Math. 6, 1–39 (1962).Google Scholar
  69. 69.
    Littlewood J.E.: On polynomials \(\sum ^{n}\pm z^{m}, \sum ^{n} e^{\alpha _{m}i} z^{m}, z= e^{\theta _{i}}\). J. Lond. Math. Soc. 41(1), 367–376 (1966).Google Scholar
  70. 70.
    Littlewood J.E.: Some Problems in Real and Complex Analysis. D. C. Heath and Co. Raytheon Education Co., Lexington (1968).Google Scholar
  71. 71.
    Logan B., Mossinghoff M.J.: Double Wieferich pairs and circulant Hadamard matrices. Preprint. http://academics.davidson.edu/math/mossinghoff/ (2015)
  72. 72.
    Maschietti A.: Difference sets and hyperovals. Des. Codes Cryptogr. 14(1), 89–98 (1998).Google Scholar
  73. 73.
    Mauduit Ch., Sárközy A.: On finite pseudorandom binary sequences. I. Measure of pseudorandomness, the Legendre symbol. Acta Arith. 82(4), 365–377 (1997).Google Scholar
  74. 74.
    Mercer I.: Merit factor of Chu sequences and best merit factor of polyphase sequences. IEEE Trans. Inf. Theory 59(9), 6083–6086 (2013).Google Scholar
  75. 75.
    Mercer I.D.: Autocorrelations of random binary sequences. Comb. Probab. Comput. 15(5), 663–671 (2006).Google Scholar
  76. 76.
    Milewski A.: Periodic sequences with optimal properties for channel estimation and fast start-up equalization. IBM J. Res. Dev. 27(5), 426–431 (1983).Google Scholar
  77. 77.
    Moon J.W., Moser L.: On the correlation function of random binary sequences. SIAM J. Appl. Math. 16(12), 340–343 (1968).Google Scholar
  78. 78.
    Mossinghoff M.J.: Wieferich pairs and Barker sequences. Des. Codes Cryptogr. 53(3), 149–163 (2009).Google Scholar
  79. 79.
    Mow W.H.: A unified construction of perfect polyphase sequences. In: IEEE International Symposium on Information Theory, p. 459. IEEE, Whistler (1995).Google Scholar
  80. 80.
    Mow W.H.: A new unified construction of perfect root-of-unity sequences. In: IEEE 4th International Symposium on Spread Spectrum Techniques and Applications, vol.3, pp. 955–959. IEEE, Mainz (1996).Google Scholar
  81. 81.
    Mow W.H., Li Sh-Y.R.: Aperiodic autocorrelation and crosscorrelation of polyphase sequences. IEEE Trans. Inf. Theory 43(3), 1000–1007 (1997).Google Scholar
  82. 82.
    Nazarathy M., Newton S.A., Giffard R.P., Moberly D.S., Sischka F., Trutna Jr. W.R., Foster S.: Real-time long range complementary correlation optical time domain reflectometer. IEEE J. Lightwave Technol. 7(1), 24–38 (1989).Google Scholar
  83. 83.
    Newman D.J.: An \(L^{1}\) extremal problem for polynomials. Proc. Am. Math. Soc. 16, 1287–1290 (1965).Google Scholar
  84. 84.
    Newman D.J., Byrnes J.S.: The \(L^{4}\) norm of a polynomial with coefficients \(\pm 1\). Am. Math. Mon. 97, 42–45 (1990).Google Scholar
  85. 85.
    No J.-S., Chung H., Yun M.-S.: Binary pseudorandom sequences of period \(2^m-1\). IEEE Trans. Inf. Theory 44(3), 1278–1282 (1998).Google Scholar
  86. 86.
    Nowicki A., Secomski W., Litniewski J., Trots I., Lewin P.A.: On the application of signal compression using Golay’s codes sequences in ultrasonic diagnostic. Arch. Acoust. 28(4), 313–324 (2003).Google Scholar
  87. 87.
    Nunn C.J., Coxson G.E.: Best-known autocorrelation peak sidelobe levels for binary codes of length 71–105. IEEE Trans. Aerosp. Electron. Syst. 44(4), 392–395 (2008).Google Scholar
  88. 88.
    Nunn C.J., Coxson G.E.: Polyphase pulse compression codes with optimal peak and integrated sidelobes. IEEE Trans. Aerosp. Electron. Syst. 45(2), 775–781 (2009).Google Scholar
  89. 89.
    Ohyama N., Honda T., Tsujiuchi J.: An advanced coded imaging without side lobes. Opt. Commun. 27(3), 339–344 (1978).Google Scholar
  90. 90.
    Paley R.E.A.C.: On orthogonal matrices. J. Math. Phys. 12, 311–320 (1933).Google Scholar
  91. 91.
    Popović B.M.: Synthesis of power efficient multitone signals with flat amplitude spectrum. IEEE Trans. Commun. 39(7), 1031–1033 (1991).Google Scholar
  92. 92.
    Rudin W.: Some theorems on Fourier coefficients. Proc. Am. Math. Soc. 10, 855–859 (1959).Google Scholar
  93. 93.
    Ryser H.J.: Combinatorial Mathematics. The Carus Mathematical Monographs, No. 14. The Mathematical Association of America; Distributed by John Wiley and Sons, Inc., New York (1963).Google Scholar
  94. 94.
    Sarwate D.V.: An upper bound on the aperiodic autocorrelation function for a maximal-length sequence. IEEE Trans. Inf. Theory IT-30(4), 685–687 (1984).Google Scholar
  95. 95.
    Schmidt B.: Cyclotomic integers and finite geometry. J. Am. Math. Soc. 12(4), 929–952 (1999).Google Scholar
  96. 96.
    Schmidt B.: Characters and cyclotomic fields in finite geometry. Lecture Notes in Mathematics, vol. 1797. Springer-Verlag, Berlin (2002).Google Scholar
  97. 97.
    Schmidt K.-U.: Binary sequences with small peak sidelobe level. IEEE Trans. Inf. Theory 58(4), 2512–2515 (2012).Google Scholar
  98. 98.
    Schmidt K.-U.: On random binary sequences. In: Sequences and Their Applications. Lecture Notes in Computer Science, vol. 7280, pp. 303–314. Springer, Berlin (2012).Google Scholar
  99. 99.
    Schmidt K.-U.: On a problem due to Littlewood concerning polynomials with unimodular coefficients. J. Fourier Anal. Appl. 19(3), 457–466 (2013).Google Scholar
  100. 100.
    Schmidt K.-U.: The peak sidelobe level of random binary sequences. Bull. Lond. Math. Soc. 46(3), 643–652 (2014).Google Scholar
  101. 101.
    Schmidt K.-U., Willms J.: Barker sequences of odd length. Des. Codes. Cryptogr. (to appear)Google Scholar
  102. 102.
    Scholtz R.A., Welch L.R.: GMW sequences. IEEE Trans. Inf. Theory 30(3), 548–553 (1984).Google Scholar
  103. 103.
    Shapiro H.S.: Extremal problems for polynomials and power series. Master’s Thesis, MIT (1951).Google Scholar
  104. 104.
    Sidelnikov V.M.: Some \(k\)-valued pseudo-random sequences and nearly equidistant codes. Probl. Inf. Transm. 5, 12–16 (1969).Google Scholar
  105. 105.
    Singer J.: A theorem in finite projective geometry and some applications to number theory. Trans. Am. Math. Soc. 43(3), 377–385 (1938).Google Scholar
  106. 106.
    Stańczak S., Boche H.: Aperiodic properties of generalized binary Rudin–Shapiro sequences and some recent results on sequences with a quadratic phase function. In Proceedings of the IEEE International Zurich Seminar on Broadband Communications, pp. 279–286 (2000).Google Scholar
  107. 107.
    Turyn R.: Optimum codes study. Technical Report, Sylvania Electronic Systems, January 1960. Final report, Contract AF19(604)-5473.Google Scholar
  108. 108.
    Turyn R.: On Barker codes of even length. Proc. IEEE 51(9), 1256–1256 (1963).Google Scholar
  109. 109.
    Turyn R.: The correlation function of a sequences of roots of 1. IEEE Trans. Inf. Theory IT-13(3), 524–525 (1967).Google Scholar
  110. 110.
    Turyn R., Storer J.: On binary sequences. Proc. Am. Math. Soc. 12(3), 394–399 (1961).Google Scholar
  111. 111.
    Turyn R.J.: Character sums and difference sets. Pac. J. Math. 15(1), 319–346 (1965).Google Scholar
  112. 112.
    Turyn R.J.: Sequences with small correlation. In: Mann H.B. (ed.) Error Correcting Codes. Wiley, New York (1968).Google Scholar
  113. 113.
    Turyn R.J.: Hadamard matrices, Baumert–Hall units, four-symbol sequences, pulse compression, and surface wave encodings. J. Comb. Theory Ser. A 16, 313–333 (1974).Google Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of PaderbornPaderbornGermany

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