Designs, Codes and Cryptography

, Volume 75, Issue 1, pp 43–57 | Cite as

Two families of nearly optimal codebooks

  • Chengju Li
  • Qin Yue
  • Yiwei Huang


Codebooks are widely applied in code-division multiple-access systems. Recently, several authors constructed codebooks meeting or nearly meeting the Welch bound (i.e. nearly optimal codebooks) using difference set, almost difference set, relative difference set, and so on. In this paper, we will give two families of nearly optimal codebooks. First, we give a class of new almost difference sets and use them to construct nearly optimal codebooks. Second, we present a general construction of codebooks from partial difference sets and obtain several classes of nearly optimal codebooks.


Signal theory Difference set Character sums 

Mathematics Subject Classification

94A12 05B10 11T24 



The paper is supported by NNSF of China (No. 11171150) and Foundation of Science and Technology on Information Assurance Laboratory (No. KJ-13-001). The authors are very grateful to the reviewers and the editor for their valuable comments and suggestions that improved the presentation and quality of this paper.


  1. 1.
    Arasu K.T., Ding C., Helleseth T., Kumar P.V., Martinsen H.: Almost difference sets and their sequences with optimal autocorrelation. IEEE Trans. Inf. Theory 47(7), 2834–2943 (2001).Google Scholar
  2. 2.
    Arasu K.T., Jungnickel D., Ma S.L., Pott A.: Strongly regular graphs with \(\lambda -\mu =1\). J. Comb. Theory Ser. A 67(1), 116–125 (1994).Google Scholar
  3. 3.
    Conway J.H., Harding R.H., Sloane N.J.A.: Packing lines, planes, etc.: packings in Grassmannian spaces. Exp. Math. 5(2), 139–159 (1996).Google Scholar
  4. 4.
    Ding C.: Complex codebooks from combinatorial designs. IEEE Trans. Inf. Theory 52(9), 4229–4235 (2006).Google Scholar
  5. 5.
    Ding C., Feng T.: A generic construction of complex codebooks meeting the Welch bound. IEEE Trans. Inf. Theory 53(11), 4245–4250 (2007).Google Scholar
  6. 6.
    Ding C., Feng T.: Codebooks from almost difference sets. Des. Codes Cryptogr. 46, 113–126 (2008).Google Scholar
  7. 7.
    Ding C., Pott A., Wang Q.: Constructions of almost difference sets from finite fieds. Des. Codes Cryptogr. (2013). doi: 10.1007/s10623-012-9789-9.
  8. 8.
    Feng T., Momihara K., Xiang Q.: Constructions of strongly regular Cayley graphs and skew Hadamard difference sets from cyclotomic classes, arXiv:1201.0701v1.Google Scholar
  9. 9.
    Feng T., Xiang Q.: Strongly regular graphs from unions of cyclotomic classess. J. Comb. Theory Ser. B 102, 982–995 (2012).Google Scholar
  10. 10.
    Fickus M., Mixon D.G., Tremain J.C.: Steiner equiangular tight frames. Linear Algebra Appl. 436, 1014–1027 (2012).Google Scholar
  11. 11.
    Hu L., Yue Q.: Gauss periods and codebooks from generalized cyclotomic sets of order four. Des. Codes Cryptogr. 69, 233–246 (2013).Google Scholar
  12. 12.
    Lidl R., Niederreiter H.: Finite Fields. Addison-Wesley Publishing Inc., London (1983).Google Scholar
  13. 13.
    Ma S.L.: Partial difference sets. Discrete Math. 52, 75–89 (1984).Google Scholar
  14. 14.
    Ma S.L.: A survey of partial difference sets. Des. Codes Cryptogr. 4, 221–261 (1994).Google Scholar
  15. 15.
    Paley R.E.A.C.: On orthogonal matrices. J. Math. Phys. 12, 311–320 (1933).Google Scholar
  16. 16.
    Sarwate D.: Meeting the Welch bound with equality. In: Ding C., Helleseth T., Niederreiter H. (eds.) Sequences and Their Applications: Proceedings of SETA’98. DMTCS Series, pp. 79–102. Springer-Verlag, New York (1999).Google Scholar
  17. 17.
    Strohmer T., Heath R.: Grassmannian frames with applications to coding and communication. Appl. Comput. Harmon. Anal. 14(3), 257–275 (2003).Google Scholar
  18. 18.
    Welch L.: Lower bounds on the maximum cross correlation of signals. IEEE Trans. Inf. Theory 20(3), 397–399 (1974).Google Scholar
  19. 19.
    Xia P., Zhou S., Giannakis G.B.: Achieving the Welch bound with difference sets. IEEE Trans. Inf. Theory 51(5), 1900–1907 (2005).Google Scholar
  20. 20.
    Yu N.Y.: A construction of codebooks associated with binary sequences. IEEE Trans. Inf. Theory 58(8) (2012).Google Scholar
  21. 21.
    Yu N.Y., Feng K., Zhang A.: A new class of near-optimal Fourier codebooks from an almost difference set. Des. Codes Cryptogr. (2012). doi: 10.1007/s10623-012-9753-8.
  22. 22.
    Zhang A., Feng K.: Construction of cyclotomic codebooks nearly meeting the Welch bound. Des. Codes Cryptogr. 63, 209–224 (2012).Google Scholar
  23. 23.
    Zhang A., Feng K.: Two classes of codebooks nearly meeting the Welch bound. IEEE Trans. Inf. Theory 58(4), 2507–2511 (2012).Google Scholar
  24. 24.
    Zhou Z., Tang X.: New nearly optimal codebooks from relative difference sets. Adv. Math. Commun. 5(3), 521–527 (2011).Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of MathematicsNanjing University of Aeronautics and AstronauticsNanjingPeople’s Republic of China
  2. 2.School of SciencesChina Pharmaceutical UniversityNanjingPeople’s Republic of China

Personalised recommendations