Skip to main content
SpringerLink
Log in

Grassmannian codes from paired difference sets

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

Abstract

An equiangular tight frame (ETF) is a sequence of vectors in a Hilbert space that achieves equality in the Welch bound and so has minimal coherence. More generally, an equichordal tight fusion frame (ECTFF) is a sequence of equi-dimensional subspaces of a Hilbert space that achieves equality in Conway, Hardin and Sloane’s simplex bound. Every ECTFF is a type of optimal Grassmannian code, that is, an optimal packing of equi-dimensional subspaces of a Hilbert space. We construct ECTFFs by exploiting new relationships between known ETFs. Harmonic ETFs equate to difference sets for finite abelian groups. We say that a difference set for such a group is “paired” with a difference set for its Pontryagin dual when the corresponding subsequence of its harmonic ETF happens to be an ETF for its span. We show that every such pair yields an ECTFF. We moreover construct an infinite family of paired difference sets using quadratic forms over the field of two elements. Together this yields two infinite families of real ECTFFs.

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. Appleby M., Bengtsson I., Dumitru I., Flammia S.: Dimension towers of SICs. I. Aligned SICs and embedded tight frames. J. Math. Phys. 58, 112201 (2017).

  2. Bachoc C., Ehler M.: Tight \(p\)-fusion frames. Appl. Comput. Harmon. Anal. 35, 1–15 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  3. Bachoc C., Bannai E., Coulangeon R.: Codes and designs in Grassmannian spaces. Discret. Math. 277, 15–28 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  4. Bajwa W.U., Calderbank R., Mixon D.G.: Two are better than one: fundamental parameters of frame coherence. Appl. Comput. Harmon. Anal. 33, 58–78 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  5. 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, 1123–1149 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  6. Barg A., Glazyrin A., Okoudjou K.A., Yu W.-H.: Finite two-distance tight frames. Linear Algebra Appl. 475, 163–175 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  7. Blokhuis A., Brehm U., Et-Taoui B.: Complex conference matrices and equi-isoclinic planes in Euclidean spaces. Beitr. Algebra Geom. 59, 491–500 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  8. Bodmann B.G.: Optimal linear transmission by loss-insensitive packet encoding. Appl. Comput. Harmon. Anal. 22, 274–285 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  9. Bodmann B.G., Elwood H.J.: Complex equiangular Parseval frames and Seidel matrices containing \(p\)th roots of unity. Proc. Am. Math. Soc. 138, 4387–4404 (2010).

    Article  MATH  Google Scholar 

  10. Bodmann B., King E.J.: Optimal arrangements of classical and quantum states with limited purity. J. Lond. Math. Soc. 101, 393–431 (2020).

  11. Brouwer A.E.: Strongly regular graphs. In: Colbourn C.J., Dinitz J.H. (eds.) Handbook of Combinatorial Designs, 2nd edn., pp. 852–868 (2007).

  12. Brouwer A.E.: Parameters of strongly regular graphs. http://www.win.tue.nl/~aeb/graphs/srg/srgtab.html

  13. 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, 436–480 (1997).

    Article  MATH  Google Scholar 

  14. Calderbank A.R., Hardin R.H., Rains E.M., Shor P.W., Sloane N.J.A.: A group-theoretic framework for the construction of packings in Grassmannian spaces. J. Algebr. Combin. 9, 129–140 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  15. Calderbank R., Thompson A., Xie Y.: On block coherence of frames. Appl. Comput. Harmon. Anal. 38, 50–71 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  16. Cameron P.J., Seidel J.J.: Quadratic forms over \(GF(2)\). Indag. Math. 76, 1–8 (1973).

    Article  MathSciNet  MATH  Google Scholar 

  17. Casazza P.G., Fickus M., Mixon D.G., Wang Y., Zhou Z.: Constructing tight fusion frames. Appl. Comput. Harmon. Anal. 30, 175–187 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  18. Cohn H., Kumar A., Minton G.: Optimal simplices and codes in projective spaces. Geom. Topol. 20, 1289–1357 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  19. Conway J.H., Hardin 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 

  20. Coutinho G., Godsil C., Shirazi H., Zhan H.: Equiangular lines and covers of the complete graph. Linear Algebra Appl. 488, 264–283 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  21. Creignou J.: Constructions of Grassmannian simplices (2008). arXiv:cs/0703036.

  22. Dhillon I.S., Heath J.R., Strohmer T., Tropp J.A.: Constructing packings in Grassmannian manifolds via alternating projection. Exp. Math. 17, 9–35 (2008).

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  24. Eldar Y.C., Kuppinger P., Bölcskei H.: Block-sparse signals: uncertainty relations and efficient recovery IEEE Trans. Signal Process. 58, 3042–3054 (2010).

    MathSciNet  MATH  Google Scholar 

  25. Et-Taoui B.: Infinite family of equi-isoclinic planes in Euclidean odd dimensional spaces and of complex symmetric conference matrices of odd orders. Linear Algebra Appl. 556, 373–380 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  26. Et-Taoui B.: Quaternionic equiangular lines. Adv. Geom. 20, 273–284 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  27. Fickus M.: Maximally equiangular frames and Gauss sums. J. Fourier Anal. Appl. 15, 413–427 (2009).

    Article  MathSciNet  MATH  Google Scholar 

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

  29. Fickus M., Schmitt C.A.: Harmonic equiangular tight frames comprised of regular simplices. Linear Algebra Appl. 586, 130–169 (2020).

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  32. Fickus M., Jasper J., King E.J., Mixon D.G.: Equiangular tight frames that contain regular simplices. Linear Algebra Appl. 555, 98–138 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  33. Fickus M., Jasper J., Mixon D.G., Peterson J.D.: Tremain equiangular tight frames. J. Combin. Theory Ser. A 153, 54–66 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  34. Fickus M., Jasper J., Mixon D.G., Peterson J.D., Watson C.E.: Equiangular tight frames with centroidal symmetry. Appl. Comput. Harmon. Anal. 44, 476–496 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  35. Fickus M., Jasper J., Mixon D.G., Peterson J.D., Watson C.E.: Polyphase equiangular tight frames and abelian generalized quadrangles. Appl. Comput. Harmon. Anal. 47, 628–661 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  36. Fickus M., Mayo B.R., Watson C.E.: Certifying the novelty of equichordal tight fusion frames (2021). arXiv:2103.03192.

  37. Fuchs C.A., Hoang M.C., Stacey B.C.: The SIC question: history and state of play. Axioms 6(21), 1–20 (2017).

    Google Scholar 

  38. Goethals J.M., Seidel J.J.: Strongly regular graphs derived from combinatorial designs. Can. J. Math. 22, 597–614 (1970).

    Article  MathSciNet  MATH  Google Scholar 

  39. Gordon D.: Difference sets. https://www.dmgordon.org/diffset/.

  40. Grove L.C.: In: Grad. Stud. Math. 39, Amer. Math. Soc., (ed.) Classical groups and geometric algebra. (2002).

  41. Hoggar S.G.: New sets of equi-isoclinic \(n\)-planes from old. Proc. Edinb. Math. Soc. 20, 287–291 (1977).

    Article  MathSciNet  MATH  Google Scholar 

  42. Holmes R.B., Paulsen V.I.: Optimal frames for erasures. Linear Algebra Appl. 377, 31–51 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  43. Iverson J.W., Mixon D.G.: Doubly transitive lines I. Higman pairs and roux. arXiv:1806.09037.

  44. Iverson J.W., Jasper J., Mixon D.G.: Optimal line packings from finite group actions. Forum Math. Sigma 8, 1–40 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  45. Jasper J., Mixon D.G., Fickus M.: Kirkman equiangular tight frames and codes. IEEE Trans. Inform. Theory. 60, 170–181 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  46. Jungnickel D., Pott A., Smith K.W.: Difference sets, in: C.J. Colbourn, J.H. Dinitz (eds.) CRC Handbook of Combinatorial Designs, pp. 419–435 (2007).

  47. King E.J.: New constructions and characterizations of flat and almost flat Grassmannian fusion frames (2016). arXiv:1612.05784.

  48. King E.J.: Creating subspace packings from other subspace packings. Linear Algebra Appl. 625, 68–80 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  49. Kocák T., Niepel M.: Families of optimal packings in real and complex Grassmannian spaces. J. Algebr. Combin. 45, 129–148 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  50. König H.: Cubature formulas on spheres. Math. Res. 107, 201–212 (1999).

    MathSciNet  MATH  Google Scholar 

  51. Kutyniok G., Pezeshki A., Calderbank R., Liu T.: Robust dimension reduction, fusion frames, and Grassmannian packings. Appl. Comput. Harmon. Anal. 26, 64–76 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  52. Lemmens P.W.H., Seidel J.J.: Equi-isoclinic subspaces of Euclidean spaces. Indag. Math. 76, 98–107 (1973).

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  55. Taylor D.E.: The Geometry of the Classical Groups. Sigma Series in Pure Mathematics 9. Heldermann Verlag (1992).

  56. Turyn R.J.: Character sums and difference sets. Pac. J. Math. 15, 319–346 (1965).

    Article  MathSciNet  MATH  Google Scholar 

  57. Waldron S.: On the construction of equiangular frames from graphs. Linear Algebra Appl. 431, 2228–2242 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  58. Waldron S.: Tight frames over the quaternions and equiangular lines. arXiv:2006.06126.

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

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  61. Zauner G.: Quantum Designs: Foundations of a Noncommutative Design Theory. Ph.D. Thesis, University of Vienna (1999).

  62. Zhang T., Ge G.: Combinatorial constructions of packings in Grassmannian spaces. Des. Codes Cryptogr. 86, 803–815 (2018).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors thank Prof. Dustin G. Mixon and the two anonymous reviewers for their thoughtful comments. We are especially grateful for the anonymous remark that led to Theorem 3.6. This work was partially supported by NSF DMS 1830066, and began during the Summer of Frame Theory (SOFT) 2016. 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.

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Air Force Institute of Technology, Wright-Patterson AFB, OH, 45433, USA

    Matthew Fickus

  2. Department of Mathematics, Iowa State University, Ames, IA, 50011, USA

    Joseph W. Iverson

  3. Department of Mathematics and Statistics, South Dakota State University, Brookings, SD, 57007, USA

    John Jasper

  4. Department of Mathematics, Colorado State University, Fort Collins, CO, 80523, USA

    Emily J. King

Authors
  1. Matthew Fickus
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Joseph W. Iverson
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. John Jasper
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Emily J. King
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Matthew Fickus.

Additional information

Communicated by K.-U. Schmidt.

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

Fickus, M., Iverson, J.W., Jasper, J. et al. Grassmannian codes from paired difference sets. Des. Codes Cryptogr. 89, 2553–2576 (2021). https://doi.org/10.1007/s10623-021-00937-w

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10623-021-00937-w

Keywords

Mathematics Subject Classification

Search

Navigation