Skip to main content
SpringerLink
Log in

Cayley sum graphs and their applications to codebooks

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

Abstract

In this paper we first provide two new constructions for Cayley sum graphs, namely, norm-coset graphs and trace-coset graphs, and determine their second largest eigenvalues using Gaussian sums. Next, a connection between Cayley sum graphs and complex codebooks is established. Based on this, infinite families of asymptotically optimal complex codebooks are explicitly constructed. The derived Cayley sum graphs and codebooks either include some known constructions as special cases or provide flexible new parameters.

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. Alon N.: Large sets in finite fields are sumsets. J. Number Theory 126, 110–118 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  2. Alon N., Rödl V.: Sharp bounds for some multicolor Ramsey numbers. Combinatorica 25, 125–141 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  3. Alon N., Rónyai L., Szabó T.: Norm-graphs: variations and applications. J. Comb. Theory Ser. B 76, 280–290 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  4. Artin M.: Algebra. Prentice Hall Inc, New Jersey (1991).

    MATH  Google Scholar 

  5. Bien F.: Constructions of telephone networks by group representations. Notices Am. Math. Soc. 36, 5–22 (1989).

    MathSciNet  MATH  Google Scholar 

  6. Brouwer A.E., Haemers W.H.: Spectra of Graphs. Springer, New York (2012).

    Book  MATH  Google Scholar 

  7. Calderbank A.R., Cameron P.J., Kantor W.M., Seidel J.J.: \({\mathbb{Z} }_4\)-kerdock codes, orthogonal spreads, and extremal Euclidean linesets. Proc. Lond. Math. Soc. 75, 436–480 (1997).

    Article  MATH  Google Scholar 

  8. Candès E.J., Wakin M.B.: An introduction to compressive sampling. IEEE Signal Process. Mag. 25, 21–30 (2008).

    Article  Google Scholar 

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

    Book  MATH  Google Scholar 

  10. Chung F.R.K.: Diameters and eigenvalues. J. Am. Math. Soc. 2, 187–196 (1989).

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  12. Crámer H.: On the distribution of primes. Proc. Cam. Philos. Soc. 20, 272–280 (1921).

    MATH  Google Scholar 

  13. Delsarte P., Goethals J.M., Seidel J.J.: Spherical codes and designs. Geometriae Dedicate 67, 363–388 (1977).

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  16. Ding C., Feng T.: Codebooks from almost difference sets. Des. Codes Cryptogr. 46, 113–126 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  17. Donoho D.L., Elad M.: Optimally sparse representation in general (nonorthogonal) dictionaries via \(\ell _1\) minimization. Proc. Natl. Acad. Sci. USA 100, 2197–2202 (2003).

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  19. Garaev M.Z.: An explicit sum-product estimate in \({\mathbb{F} }_q\). Int. Math. Res. Not. 2007, 1–11 (2007).

    Google Scholar 

  20. Golomb S.W., Gong G.: Signal Design for Good Correlation: For Wireless Communication. Cryptography and Radar. Cambridge University Press, Cambridge (2005).

    Book  MATH  Google Scholar 

  21. Grynkiewicz D., Lev V.F., Serra O.: The connectivity of addition Cayley graphs. Electron. Notes Discret. Math. 29, 135–139 (2007).

    Article  MATH  Google Scholar 

  22. He X., Wigderson Y.: Multicolor Ramsey numbers via pseudorandom graphs. Electron. J. Comb. 27, 32 (2020).

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  24. Hoory S., Linial N., Wigderson A.: Expander graphs and their applications. Bull. Am. Math. Soc. (N.S.) 43, 439–561 (2006).

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  26. Krivelevich M., Sudakov B.: Pseudo-random graphs. In: Györi E., Katona G.O.H., Lovász L., Fleiner T. (eds.) More Sets, Graphs and Numbers. Bolyai Society Mathematical Studies, vol. 15. Springer, Berlin (2006).

    Google Scholar 

  27. Lenz J., Mubayi D.: Multicolor Ramsey numbers for complete bipartite versus complete graphs. J. Graph Theory 77, 19–38 (2014).

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  29. Li W.-C.W., Feng K.Q.: Character sums and abelian Ramanujan graphs. J. Number Theory 41, 199–217 (1992).

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  31. Li S., Ge G.: Deterministic construction of sparse sensing matrices via finite geometry. IEEE Trans. Signal Process. 62, 2850–2859 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  32. Li S., Gao F., Ge G., Zhang S.: Deterministic construction of compressed sensing matrices via algebraic curves. IEEE Trans. Inf. Theory 58, 5035–5041 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  33. Lidl R., Niederreiter H.: Finite Fields. Cambridge University Press, Cambridge (1984).

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  36. Massey J.L., Mittelholzer T.: Welchs bound and sequence sets for code-division multiple-access systems. In: Capocelli R., De Santis A., Vaccaro U. (eds.) Sequences II. Springer, New York (1993).

    Google Scholar 

  37. Mubayi D., Verstraëte J.: A note on pseudorandom Ramsey graphs. arXiv:1909.01461

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

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  40. Sarwate D.V.: Meeting the Welch bound with equality. In: Ding C., Helleseth T., Niederreiter H. (eds.) Sequences and their Applications. Discrete Mathematics and Theoretical Computer Science. Springer, London (1999).

    Google Scholar 

  41. Satake S.: Certain codebooks and the generalized Erdős–Falconer distance problem (extended abstract). In: Sequences and their Applications (2020).

  42. Satake S.: On the restricted isometry property of the Paley matrix. Linear Algebra Appl. 631, 35–47 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  43. Satake S., Gu Y.: Constructions of complex codebooks asymptotically meeting the Welch bound: A graph theoretic approach. In: 2020 IEEE International Symposium on Information Theory (pp. 48–53) (2020).

  44. Satake S., Gu Y.: On compressed sensing matrices breaking the square-root bottleneck. In: 2020 IEEE Information Theory Workshop (pp. 1–5) (2020).

  45. Sipser M., Spielman D.: Expander codes. IEEE Trans. Inf. Theory 42, 1710–1722 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  46. Solymosi J.: Incidences and the spectra of graphs. In: Combinatorial Number Theory and Additive Group Theory. Advanced Courses in Mathematics - CRM Barcelona. Birkhäuser Basel (2009).

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

    Article  MathSciNet  MATH  Google Scholar 

  48. Szabó T.: On the spectrum of projective norm-graphs. Inform. Process. Lett. 86, 71–74 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  49. Tan P., Zhou Z., Zhang D.: A construction of codebooks nearly achieving the Levenstein bound. IEEE Signal Process. Lett. 23, 1306–1309 (2016).

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  51. Wootters W., Fields B.: Optimal state-determination by mutually unbiased measurements. Ann. Phys. 191, 363–381 (1989).

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  53. Xu G., Xu Z.: Compressed sensing matrices from Fourier matrices. IEEE Trans. Inf. Theory 61, 469–478 (2015).

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  55. Zémor G.: Hash functions and Cayley graphs. Des. Codes Cryptogr. 4, 381–394 (1994).

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  59. Zhou Z., Ding C., Li N.: New families of codebooks achieving the Levenshtein bound. IEEE Trans. Inf. Theory 60, 7382–7387 (2014).

    Article  MATH  Google Scholar 

Download references

Acknowledgements

The authors are very grateful to the anonymous reviewers and associated editor for their insightful and constructive comments and suggestions. The authors would like to thank Professor Ofer Shayevitz and Professor Rami Zamir for helpful discussions. S. Satake has been supported by Grant-in-Aid for JSPS Fellows 18J11282 and 20J00469 of the Japan Society for the Promotion of Science and ACT-X JPMJAX2109 of the Japan Science and Technology Agency. Y. Gu has been supported by Grant-in-Aid for Early-Career Scientists 21K13830 of the Japan Society for the Promotion of Science. Parts of this work were presented at [43].

Author information

Authors and Affiliations

  1. School of Interdisciplinary Mathematical Sciences, Meiji University, Tokyo, 164-8525, Japan

    Shohei Satake

  2. Faculty of Information Science and Electrical Engineering, Kyushu University, Fukuoka, 819-0395, Japan

    Yujie Gu

Authors
  1. Shohei Satake
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yujie Gu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yujie Gu.

Additional information

Communicated by T. Helleseth.

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

Satake, S., Gu, Y. Cayley sum graphs and their applications to codebooks. Des. Codes Cryptogr. 91, 1315–1333 (2023). https://doi.org/10.1007/s10623-022-01152-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10623-022-01152-x

Keywords

Mathematics Subject Classification

Search

Navigation