Skip to main content
SpringerLink
Log in

Jacobi sums over Galois rings of arbitrary characters and their applications in constructing asymptotically optimal codebooks

  • Published:
Designs, Codes and Cryptography Aims and scope Submit manuscript

Abstract

Codebooks with small maximum cross-correlation amplitudes are used to distinguish the signals from different users in multiple access communication systems in code division. Firstly, this paper studies the Jacobi sums over Galois rings of arbitrary characteristics and completely determines their absolute values. This extends the work by Li et al. (Sci China 56(7):1457–1465, 2013), where the Jacobi sums over some Galois rings with characteristics of a prime square were discussed. It is worth mentioning that the generalization is not trivial, as the Galois rings of arbitrary characteristics have a more complicated structure. Our deterministic construction of codebooks is based on Jacobi sums over Galois rings of arbitrary characteristics, producing asymptotically optimal codebooks for the Welch bound. Finally, compared to the literature, this article proposes for the first time design of codebooks based on Jacobi sums over Galois rings. In addition, the parameters of the presented 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. Calderbank A.R., Cameron P.J., Kantor W.M., Seidel J.J.: \(\mathbb{Z} _4\)-Kerdock codes, orthogonal spreads, and extremal Euclidean line sets. Proc. Lond. Math. Soc. 75(3), 436–480 (1997).

    Google Scholar 

  2. Carlet C.: One-weight \(\mathbb{z}_4\)-linear codes. In: Proc. of an International Conference on Coding Theory Cryptography and Related Areas, Guanajuato, Mexico, pp. 57–72 (2000).

  3. Christensen O.: An Introduction to Frames and Riesz Bases. Birkhäuser, Boston (2003).

    Google Scholar 

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

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  6. Delsarte P., Goethals J.M., Seidel J.J.: Bounds for systems of lines and Jacobi polynomials. Philips Res. Repts. 30, 91–105 (1975).

    Google Scholar 

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

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  9. Feng T.: A new construction of perfect nonlinear functions using Galois rings. J. Comb. Des. 17(3), 229–239 (2009).

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  12. Hammons A.R., Kumar P.V., Calderbank A.R., Sloane N.J.A., Solé P.: The \(\mathbb{Z} _4\)-linearity of Kerdock, Preparata, Goethals, and related codes. IEEE Trans. Inf. Theory 40(12), 301–319 (1994).

    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).

    MathSciNet  Google Scholar 

  14. Hong S., Park H., No J., 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).

    MathSciNet  Google Scholar 

  15. Hou X., Leung K., Xiang Q.: New partial difference sets in \(\mathbb{Z} _{p^2}^t\) and a related problem about Galois rings. Finite Fields Appl. 7(1), 165–188 (2001).

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  17. Kabatyanskii G.A., Levenshtein V.I.: Bounds for packing on a sphere and in space. Probl. Inf. Transm. 14, 1–17 (1978).

    MathSciNet  Google Scholar 

  18. Langevin P., Solé P.: Gauss sums over quasi-Frobenius rings. In: Proc. of the Fifth International Conference on Finite Fields and Applications, pp. 329–340, (1999).

  19. Levenshtein V.I.: Bounds for packings of metric spaces and some of their applications. Probl. Cybern. 40, 43–110 (1983) (in Russian).

  20. Li S., Ge G.: Deterministic sensing matrices arising from near orthogonal systems. IEEE Trans. Inf. Theory 60(4), 2291–2302 (2014).

    MathSciNet  Google Scholar 

  21. Li C., Yue Q., Huang Y.: Two families of nearly optimal codebooks. Des. Codes Cryptogr. 75(1), 43–57 (2015).

    MathSciNet  Google Scholar 

  22. Li J., Zhu S., Feng K.: The Gauss sums and Jacobi sums over Galois ring \(GR(p^2, r)\). Sci. China 56(7), 1457–1465 (2013).

    MathSciNet  Google Scholar 

  23. Lidl R., Niederreiter H.: Finite Fields and Their Applications, 2nd edn Cambridge University Press, Cambridge (1997).

    Google Scholar 

  24. Love D.J., Heath R.W., Strohmer T.: Grassmannian beamforming for multiple input multiple output wireless systems. IEEE Trans. Inf. Theory 49(10), 2735–2747 (2003).

    MathSciNet  Google Scholar 

  25. Lu W., Wu X., Cao X., Chen M.: Six constructions of asymptotically optimal codebooks via the character sums. Des. Codes Cryptogr. 88(2), 1139–1158 (2020).

    MathSciNet  Google Scholar 

  26. Luo G., Cao X.: Two constructions of asymptotically optimal codebooks via the hyper Eisenstein sum. IEEE Trans. Inf. Theory 64(10), 6498–6505 (2017).

    MathSciNet  Google Scholar 

  27. Luo G., Cao X.: New constructions of codebooks asymptotically achieving the Welch bound. In: IEEE Int. Symp. Inf. Theory, Vail, CO, USA, pp. 2346–2350 (2018).

  28. Luo G., Cao X.: Two constructions of asymptotically optimal codebooks. Cryptogr. Commun. 11(2), 825–838 (2019).

    MathSciNet  Google Scholar 

  29. Massey J., Mittelholzer T.: Welch\(^{,}\)s bound and sequence sets for code-division multiple-access systems. Sequences II: Methods in Communication, Security and Computer Science, pp. 63–78. Springer, New York (1999).

  30. Qian L., Cao X.: Gaussian sums, hyper Eisenstein sums, and Jacobi sums over a local ring and their applications. AAECC 34(2), 211–244 (2021).

    MathSciNet  Google Scholar 

  31. Qi Y., Mesnager S., Tang C.: Codebooks from generalized bent \(\mathbb{Z} _4\)-valued quadratic forms. Discret. Math. 343(3), 111736 (2020).

    Google Scholar 

  32. Sarwate D.: Meeting the Welch bound with equality. In: Proc. SETA\(^{\prime }\)\(98\), pp. 79–102. Springer, Berlin (1999).

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

    MathSciNet  Google Scholar 

  34. Tarokh V., Kim I.M.: Existence and construction of noncoherent unitary space-time codes. IEEE Trans. Inf. Theory 48(12), 3112–3117 (2002).

    MathSciNet  Google Scholar 

  35. Tian L., Li Y., Liu T., Xu C.: Constructions of codebooks asymptotically achieving the Welch bound with additive characters. IEEE Signal Process. Let. 26(4), 622–626 (2019).

    Google Scholar 

  36. Wan Z.: Lectures Notes on Finite Fields and Galois Rings. Word Scientific, Singapore (2003).

    Google Scholar 

  37. Wang J.: On the Jacobi sums modulo \(p^n\). J. Number Theory 39(1), 50–64 (1991).

    MathSciNet  Google Scholar 

  38. Wang Q., Yan Y.: Asymptotically optimal codebooks derived from generalized bent functions. IEEE Access 8, 54905–54909 (2020).

    Google Scholar 

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

    Google Scholar 

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

    MathSciNet  Google Scholar 

  41. Xiang Q., Davis J.A.: Constructions of low-rank relative differences sets in \(2\)-groups using Galois rings. Finite Fields Appl. 6(2), 130–145 (2000).

    MathSciNet  Google Scholar 

  42. Xu D.: Construction of mutually unbiased maximally entangled bases in \(\mathbb{C} ^{2^s}\otimes \mathbb{C} ^{2^s}\) by using Galois rings. Quantum Inf. Process. 19(6), 175 (2020).

    Google Scholar 

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

    MathSciNet  Google Scholar 

  44. Yu N.Y., Feng K., Zhang A.X.: A new class of near-optimal partial Fourier codebooks from an almost difference set. Des. Codes Cryptogr. 71(3), 493–501 (2014).

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  46. Zhang A.X., Feng K.: Construction of cyclotomic codebooks nearly meeting the Welch bound. Des. Codes Cryptogr. 63(2), 209–224 (2012).

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors thank the Associated Editor and anonymous reviewers for their valuable comments that improved the presentation and quality of this paper. G. Wang and Y. Gao are supported by the Scientific Research Project of Tianjin Education Commission (No. 2022KJ075), the Fundamental Research Funds for the Central Universities of China (No. 3122023QD25) and the Open Foundation of Tianjin Key Laboratory of Advanced Signal Processing (No. 2022ASP-TJ02). The French Agence Nationale de la Recherche partially supported the third author’s work through ANR BARRACUDA (ANR-21-CE39-0009). F.-W Fu is supported by the National Key Research and Development Program of China (Grant No. 2018YFA0704703), the National Natural Science Foundation of China (Grant No. 61971243), the Natural Science Foundation of Tianjin (20JCZDJC00610), the Fundamental Research Funds for the Central Universities of China (Nankai University), and the Nankai Zhide Foundation.

Author information

Authors and Affiliations

  1. Sino-European Institute of Aviation Engineering, Civil Aviation University of China, Tianjin, 300300, China

    Deng-Ming Xu

  2. College of Science, Civil Aviation University of China, Tianjin, 300300, China

    Gang Wang & You Gao

  3. Department of Mathematics, University of Paris VIII, 93526, Saint-Denis, France

    Sihem Mesnager

  4. Laboratory Geometry, Analysis and Applications, LAGA, University Sorbonne Paris Nord, CNRS, UMR 7539, 93430, Villetaneuse, France

    Sihem Mesnager

  5. Telecom Paris, Polytechnic Institute of Paris, 91120, Palaiseau, France

    Sihem Mesnager

  6. Chern Institute of Mathematics and LPMC, Nankai University, Tianjin, 300071, China

    Fang-Wei Fu

Authors
  1. Deng-Ming Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gang Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sihem Mesnager
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. You Gao
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Fang-Wei Fu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sihem Mesnager.

Additional information

Communicated by G. McGuire.

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xu, DM., Wang, G., Mesnager, S. et al. Jacobi sums over Galois rings of arbitrary characters and their applications in constructing asymptotically optimal codebooks. Des. Codes Cryptogr. 92, 1051–1073 (2024). https://doi.org/10.1007/s10623-023-01328-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10623-023-01328-z

Keywords

Mathematics Subject Classification

Search

Navigation