Optimal Line Packings from Nonabelian Groups

Abstract

We use group schemes to construct optimal packings of lines through the origin. In this setting, optimal line packings are naturally characterized using representation theory, which in turn leads to a necessary integrality condition for the existence of equiangular central group frames. We conclude with an infinite family of optimal line packings using the group schemes associated with certain Suzuki 2-groups, specifically, extensions of Heisenberg groups. Notably, this is the first known infinite family of equiangular tight frames generated by representations of nonabelian groups.

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References

  1. 1.

    Bandeira, A.S., Fickus, M., Mixon, D.G., Wong, P.: The road to deterministic matrices with the restricted isometry property. J. Fourier Anal. Appl. 19(6), 1123–1149 (2013). https://doi.org/10.1007/s00041-013-9293-2

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Bannai, E.: Association schemes and fusion algebras (an introduction). J. Algebr. Comb. 2(4), 327–344 (1993). https://doi.org/10.1023/A:1022489416433

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Bannai, E., Hoggar, S.G.: On tight \(t\)-designs in compact symmetric spaces of rank one. In: Proceedings of the Japan Academy Series A, Mathematical Sciences 61(3), 78–82 (1985). http://projecteuclid.org/euclid.pja/1195514811

    MathSciNet  Article  Google Scholar 

  4. 4.

    Bannai, E., Ito, T.: Algebraic Combinatorics I: Association Schemes. Benjamin, Menlo Park (1984)

    Google Scholar 

  5. 5.

    Barg, A., Glazyrin, A., Okoudjou, K.A., Yu, W.-H.: Finite two-distance tight frames. Linear Algebra Appl. 475, 163–175 (2015). https://doi.org/10.1016/j.laa.2015.02.020

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    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. London Math. Soc. 75(2), 436–480 (1997)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Conway, J.H., Hardin, R.H., Sloane, N.J.A.: Packing lines, planes, etc.: packings in Grassmannian spaces. Experiment. Math. 5(2), 139–159 (1996). http://projecteuclid.org/euclid.em/1047565645

    MathSciNet  Article  Google Scholar 

  8. 8.

    Daubechies, I., Grossmann, A., Meyer, Y.: Painless nonorthogonal expansions. J. Math. Phys. 27(5), 1271–1283 (1986). https://doi.org/10.1063/1.527388

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Delsarte, P.: An algebraic approach to the association schemes of coding theory. Philips Res. Rep. Suppl. (10), vi+97 (1973)

  10. 10.

    Delsarte, P., Goethals, J.M., Seidel, J.J.: Spherical codes and designs. Geom. Dedicata 6(3), 363–388 (1977)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Ding, C., Feng, T.: A generic construction of complex codebooks meeting the Welch bound. IEEE Trans. Inform. Theory 53(11), 4245–4250 (2007). https://doi.org/10.1109/TIT.2007.907343

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Duffin, R.J., Schaeffer, A.C.: A class of nonharmonic Fourier series. Trans. Amer. Math. Soc. 72, 341–366 (1952)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Fickus, M., Jasper, J., Mixon, D.G., Peterson, J.D., Watson, C.E.: Equiangular tight frames with centroidal symmetry. Appl. Comput. Harmon. Anal. 44(2), 476–496 (2018). https://doi.org/10.1016/j.acha.2016.06.004

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Fickus, M., Mixon, D.G.: Tables of the existence of equiangular tight frames (2015). arXiv:1504.00253

  15. 15.

    The GAP Group: GAP—Groups, Algorithms, and Programming, Version 4.8.4 (2016). http://www.gap-system.org

  16. 16.

    Godsil, C.: Association schemes (06-03-2010). http://www.math.uwaterloo.ca/~cgodsil/pdfs/assoc2.pdf

  17. 17.

    Goyal, V.K., Kovačević, J., Kelner, J.A.: Quantized frame expansions with erasures. Appl. Comput. Harmon. Anal. 10(3), 203–233 (2001). https://doi.org/10.1006/acha.2000.0340

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    Hanaki, A.: A condition on lengths of conjugacy classes and character degrees. Osaka J. Math. 33(1), 207–216 (1996). URL http://projecteuclid.org/euclid.ojm/1200786698

  19. 19.

    Hanaki, A., Okuyama, T.: Groups with some combinatorial properties. Osaka J. Math. 34(2), 337–356 (1997). http://projecteuclid.org/euclid.ojm/1200787498

  20. 20.

    Higman, G.: Suzuki \(2\)-groups. Ill. J. Math. 7, 79–96 (1963)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Hochwald, B.M., Marzetta, T.L., Richardson, T.J., Sweldens, W., Urbanke, R.: Systematic design of unitary space-time constellations. IEEE Trans. Inform. Theory 46(6), 1962–1973 (2000)

    Article  Google Scholar 

  22. 22.

    Holmes, R.B., Paulsen, V.I.: Optimal frames for erasures. Linear Algebra Appl. 377, 31–51 (2004). https://doi.org/10.1016/j.laa.2003.07.012

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    Holt, D.F., Eick, B., O’Brien, E.A.: Handbook of Computational Group Theory. Discrete Mathematics and Its Applications (Boca Raton). Chapman & Hall/CRC, Boca Raton (2005). https://doi.org/10.1201/9781420035216

    Google Scholar 

  24. 24.

    Iverson, J.W.: Frames generated by compact group actions. Trans. Amer. Math. Soc. 370(1), 509–551 (2018). https://doi.org/10.1090/tran/6954

    MathSciNet  Article  MATH  Google Scholar 

  25. 25.

    Iverson, J.W., Jasper, J., Mixon, D.G.: Central group frames (2016). https://github.com/jwiverson/central-group-frames

  26. 26.

    Iverson, J.W., Jasper, J., Mixon, D.G.: Optimal line packings from finite group actions (2017). arXiv:1709.03558

  27. 27.

    Iverson, J.W., Mixon, D.G.: Doubly transitive lines i: Higman pairs and roux (2018). arXiv:1806.09037

  28. 28.

    James, G., Liebeck, M.: Representations and Characters of Groups, 2nd edn. Cambridge University Press, New York (2001). https://doi.org/10.1017/CBO9780511814532

  29. 29.

    Kabatiansky, G.A., Levenshtein, V.I.: On bounds for packings on a sphere and in space. Probl. Peredachi Inf. 14(1), 3–25 (1978)

    MathSciNet  Google Scholar 

  30. 30.

    Lang, S.: Algebra. Revised third edition. Springer, New York (2002)

    Google Scholar 

  31. 31.

    Lempel, A.: Matrix factorization over \({\rm GF}(2)\) and trace-orthogonal bases of \({\rm GF}(2^{n})\). SIAM J. Comput. 4, 175–186 (1975)

    MathSciNet  Article  Google Scholar 

  32. 32.

    Lempel, A., Weinberger, M.J.: Self-complementary normal bases in finite fields. SIAM J. Discrete Math. 1(2), 193–198 (1988). https://doi.org/10.1137/0401021

    MathSciNet  Article  MATH  Google Scholar 

  33. 33.

    Lidl, R., Niederreiter, H.: Finite Fields. Encyclopedia of Mathematics and its Applications. Advanced Book Program, vol. 20. Addison-Wesley, Reading (1983)

  34. 34.

    Lübeck, F.: FUtil, Version 0.1.5 (2017). http://www.math.rwth-aachen.de/~Frank.Luebeck/gap/FUtil/

  35. 35.

    Newton, I.: In: Turnball, H.W. (ed.) The Correspondence of Isaac Newton, vol. 3, pp. 1688–1694. Cambridge University Press, Cambridge (1966)

  36. 36.

    Prasad, A.: On character values and decomposition of the Weil representation associated to a finite abelian group. J. Anal. 17, 73–85 (2009)

    MathSciNet  MATH  Google Scholar 

  37. 37.

    Rankin, R.A.: On the minimal points of positive definite quadratic forms. Mathematika 3, 15–24 (1956)

    MathSciNet  Article  Google Scholar 

  38. 38.

    Renes, J.M.: Equiangular tight frames from Paley tournaments. Linear Algebra Appl. 426(2–3), 497–501 (2007)

    MathSciNet  Article  Google Scholar 

  39. 39.

    Roy, A.: Complex Lines with Restricted Angles. ProQuest LLC, Ann Arbor, MI (2006). http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_v al_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss :NR17475. PhD thesis, University of Waterloo (Canada)

  40. 40.

    The Sage Developers: SageMath, the Sage Mathematics Software System, Version 7.3 (2016). http://www.sagemath.org

  41. 41.

    Schläfli, L.: Theorie der vielfachen Kontinuität. Denkschriften der schweizerischen Naturforschenden Gesellschaft, Birkhäuser (1901)

    Google Scholar 

  42. 42.

    Scott, A.J., Grassl, M.: Symmetric informationally complete positive-operator-valued measures: a new computer study. J. Math. Phys. 51(4), 042203 (2010). https://doi.org/10.1063/1.3374022

    MathSciNet  Article  MATH  Google Scholar 

  43. 43.

    Seroussi, G., Lempel, A.: Factorization of symmetric matrices and trace-orthogonal bases in finite fields. SIAM J. Comput. 9(4), 758–767 (1980). https://doi.org/10.1137/0209059

    MathSciNet  Article  MATH  Google Scholar 

  44. 44.

    Strohmer, T.: A note on equiangular tight frames. Linear Algebra Appl. 429(1), 326–330 (2008)

    MathSciNet  Article  Google Scholar 

  45. 45.

    Strohmer, T., Heath Jr., R.W.: Grassmannian frames with applications to coding and communication. Appl. Comput. Harmon. Anal. 14(3), 257–275 (2003). https://doi.org/10.1016/S1063-5203(03)00023-X

    MathSciNet  Article  MATH  Google Scholar 

  46. 46.

    Tammes, P.M.L.: On the origin of number and arrangement of the places of exit on the surface of pollen grains. Recueil de Travaux Botaniques Néerlandais 27, 1–84 (1930)

    Google Scholar 

  47. 47.

    Thill, M., Hassibi, B.: Low-coherence frames from group Fourier matrices. IEEE Trans. Inform. Theory 63(6), 3386–3404 (2017). https://doi.org/10.1109/TIT.2017.2686420

    MathSciNet  Article  MATH  Google Scholar 

  48. 48.

    Vale, R., Waldron, S.: Tight frames generated by finite nonabelian groups. Numer. Algorithms 48(1–3), 11–27 (2008). https://doi.org/10.1007/s11075-008-9167-x

    MathSciNet  Article  MATH  Google Scholar 

  49. 49.

    Wang, C.C.: An algorithm to design finite field multipliers using a self-dual normal basis. IEEE Trans. Comput. 38(10), 1457–1460 (1989)

    Article  Google Scholar 

  50. 50.

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

    Article  Google Scholar 

  51. 51.

    Wildberger, N.J.: Duality and entropy of finite commutative hypergroups and fusion rule algebras. J. London Math. Soc. 56(2), 275–291 (1997)

    MathSciNet  Article  Google Scholar 

  52. 52.

    Xia, P., Zhou, S., Giannakis, G.B.: Achieving the Welch bound with difference sets. IEEE Trans. Inform. Theory 51(5), 1900–1907 (2005). https://doi.org/10.1109/TIT.2005.846411

    MathSciNet  Article  MATH  Google Scholar 

  53. 53.

    Zauner, G.: Quantendesigns: Grundzüge einer nichtkommutativen Designtheorie. Ph.D. thesis, University of Vienna (1999)

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Acknowledgements

The authors thank the anonymous referees for thoughtful recommendations that significantly altered and greatly improved the manuscript. Part of this work was conducted during the SOFT 2016: Summer of Frame Theory workshop at the Air Force Institute of Technology. The authors thank Nathaniel Hammen for helpful discussions during this workshop. This work was partially supported by NSF DMS 1321779, ARO W911NF-16-1-0008, AFOSR F4FGA05076J002, an AFOSR Young Investigator Research Program award, and an AFRL Summer Faculty Fellowship Program award. The views expressed in this article are those of the authors and do not reflect the official policy or position of the United States Air Force, Department of Defense, or the U.S. Government.

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Correspondence to Joseph W. Iverson.

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Iverson, J.W., Jasper, J. & Mixon, D.G. Optimal Line Packings from Nonabelian Groups. Discrete Comput Geom 63, 731–763 (2020). https://doi.org/10.1007/s00454-019-00084-z

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Keywords

  • Equiangular tight frames
  • Association schemes
  • Difference sets
  • Group frames
  • Heisenberg group

Mathematics Subject Classification

  • Primary: 05B10
  • 42C15
  • 94C30
  • Secondary: 20C15
  • 52C99