Skip to main content
SpringerLink
Log in

Two constructions of asymptotically optimal codebooks via the trace functions

  • Published:
Cryptography and Communications Aims and scope Submit manuscript

Abstract

In this paper, we present two new constructions of complex codebooks with multiplicative characters, additive characters and trace functions over finite fields, and determin the maximal cross-correlation amplitude of these codebooks. We prove that the codebooks we constructed are asymptotically optimal with respect to the Welch bound. Moreover, in the first construction, we generalize the result in Zhang and Feng (IEEE Trans. Inform. Theory 58(4), 2507–2511, 2012). In the second construction, we generalize the results in Hong et al. (IEEE Trans. Inf. Theory 60(6), 3698–3705, 2014), we can asymptotically achieve Welch bound for any odd prime p, we also derive the whole distribution of their inner products. The parameters of these codebooks are new.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Candes, E., Wakin, M.: An introduction to compressive sampling. IEEE Signal Process 25(2), 21–30 (2008)

    Article  Google Scholar 

  2. Conway, J., Harding, R., Sloane, N.: Packing lines, planes, etc.: Packings in Grassmannian spaces. Exp. Math. 5(2), 139–159 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ding, C.: Complex codebooks from combinatorial designs. IEEE Trans. Inform. Theory 52(9), 4229–4235 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ding, C., Feng, T.: A generic construction of complex codebooks meeting the Welch bound. IEEE Trans. Inf. Theory 53(11), 4245–4250 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Delsarte, P., Goethals, J., Seidel, J.: Spherical codes and designs. Geometriae Dedicate 67(3), 363–388 (1997)

    MathSciNet  MATH  Google Scholar 

  6. Fickus, M., Mixon, D., Tremain, J.: Steiner equiangular tight frames. Linear Algebra Appl. 436(5), 1014–1027 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. Fickus, M., Mixon, D.: Tables of the existence of equiangular tight frames, arXiv:1504.00253v2 (2016)

  8. Fickus, M., Mixon, D., Jasper, J.: Equiangular tight frames from hyperovals. IEEE Trans. Inf. Theory 62(9), 5225–5236 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  9. Fickus, M., Jasper, J., Mixon, D., Peterson, J.: Tremain equiangular tight frames, arXiv:1602.03490v1 (2016)

  10. Helleseth, T., Kholosha, A.: Monomial and quadratic bent functions over the finite fields of odd characteristic. IEEE Trans. Inf. Theory 52(5), 2018–2032 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hu, H., Wu, J.: New constructions of codebooks nearly meeting the Welch bound with equality. IEEE Trans. Inf. Theory 60(2), 1348–1355 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hong, S., Park, H., Helleseth, T., Kim, Y.: Near optimal partial Hadamard codebook construction using binary sequences obtained from quadratic residue mapping. IEEE Trans. Inf. Theory 60(6), 3698–3705 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. Heng, Z., Ding, C., Yue, Q.: New constructions of asymptotically optimal codebooks with multiplicative characters. IEEE Trans. Inf. Theory 63(10), 6179–6187 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  14. Heng, Z: Nearly optimal codebooks based on generalized Jacobi sums. Discrete Appl. Math. 250, 227–240 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kovacevic, J., Chebira, A.: An introduction to frames. Found. Trends Signal Process. 2(1), 1–94 (2008)

    Article  MATH  Google Scholar 

  16. Li, C.J., Qin, Y., Huang, Y.W.: Two families of nearly optimal codebooks. Des. Codes Cryptogr. 75(1), 43–57 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  17. Lu, W., Wu, X., Cao, X.W., Chen, M.: Six constructions of asymptotically optimal codebooks via the character sums. Des. Codes Cryptogr. 88(6), 1139–1158 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  18. Luo, G.J., Cao, X.W.: Two constructions of asymptotically optimal codebooks via the hyper Eisenstein sum. IEEE Trans. Inf. Theory 64(10), 6498–6505 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  19. Luo, G.J., Cao, X.W.: New constructions of codebooks asymptotically achieving the Welch bound. In: Proceedings IEEE Int. Symp. Inf. Theory, Vail, CO USA, pp 2346–2349 (2018)

  20. Luo, G.J., Cao, X.W.: Two constructions of asymptotically optimal codebooks, Crypt. Commun. [online]. Available: https://doi.org/10.1007/s12095-018-0331-4 (2018)

  21. Lidl, R., Niederreiter, H.: Finite fields. Cambridge University Press, Cambridge (1997)

    MATH  Google Scholar 

  22. Massey, J., Mittelholzer, T.: Welch’s bound and sequence sets for code-division multiple-access systems, Sequences II, pp 63–78. Springer, New York, (1999)

    MATH  Google Scholar 

  23. Renes, J., Blume-Kohout, R., Scot, A., Caves, C.: Symmetric informationally complete quantum measurements. J. Math. Phys. 45(6), 2171–2180 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  24. Rahimi, F.: Covering graphs and equiangular tight frames, Ph.D. Thesis, University of Waterloo, Ontario, (available at http://hdl.handle.net/10012/10793) (2016)

  25. Sarwate, D.: Meeting the Welch bound with equality, pp 63–79. Springe, New York (1999)

    MATH  Google Scholar 

  26. Strohmer, T., Heath, R.: Grassmannian frames with applications to coding and communication. Appl. Comput. Harmon. Anal. 14(3), 257–275 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  27. Welch, L.: Lower bounds on the maximum cross correlation of signals. IEEE Trans. Inform. Theory 20(3), 397–399 (1974)

    Article  MATH  Google Scholar 

  28. Xia, P., Zhou, S., Giannakis, G.: Achieving the Welch bound with difference sets. IEEE Trans. Inform. Theory 51(5), 1900–1907 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  29. Yu, N.: A construction of codebooks associated with binary sequences. IEEE Trans. Inform. Theory 58(8), 5522–5533 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  30. Zhang, A., Feng, K.: Two classes of codebooks nearly meeting the Welch bound. IEEE Trans. Inform. Theory 58(4), 2507–2511 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  31. Zhang, A., Feng, K.: Construction of cyclotomic codebooks nearly meeting the Welch bound. Des. Codes Cryptogr. 63(2), 209–224 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  32. Zhou, Z., Tang, X.: New nearly optimal codebooks from relative difference sets. Adv. Math. Commun. 5(3), 521–527 (2011)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant No. 11971102, 11801070, 11771007, 61572027) and the Basic Research Foundation (Natural Science).

Author information

Authors and Affiliations

  1. School of Mathematics, Southeast University, Nanjing, 210096, China

    Xia Wu & Wei Lu

  2. Department of Math, Nanjing University of Aeronautics and Astronautics, Nanjing, 211100, China

    Xiwang Cao

Authors
  1. Xia Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wei Lu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xiwang Cao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wei Lu.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wu, X., Lu, W. & Cao, X. Two constructions of asymptotically optimal codebooks via the trace functions. Cryptogr. Commun. 12, 1195–1211 (2020). https://doi.org/10.1007/s12095-020-00448-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12095-020-00448-w

Keywords

Mathematics Subject Classification (2010)

Search

Navigation