Bounds for codes and designs in complex subspaces



We introduce the concepts of complex Grassmannian codes and designs. Let \(\mathcal{G}_{m,n}\) denote the set of m-dimensional subspaces of ℂn: then a code is a finite subset of \(\mathcal{G}_{m,n}\) in which few distances occur, while a design is a finite subset of \(\mathcal{G}_{m,n}\) that polynomially approximates the entire set. Using Delsarte’s linear programming techniques, we find upper bounds for the size of a code and lower bounds for the size of a design, and we show that association schemes can occur when the bounds are tight. These results are motivated by the bounds for real subspaces recently found by Bachoc, Bannai, Coulangeon and Nebe, and the bounds generalize those of Delsarte, Goethals and Seidel for codes and designs on the complex unit sphere.


Codes Designs Bounds Grassmannian spaces Complex subspaces Linear programming Delsarte Association schemes 


  1. 1.
    Agrawal, D., Richardson, T.J., Urbanke, R.L.: Multiple-antenna signal constellations for fading channels. IEEE Trans. Inf. Theory 47, 2618–2626 (2001) MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bachoc, C.: Linear programming bounds for codes in Grassmannian spaces. IEEE Trans. Inf. Theory 52, 2111–2125 (2006) CrossRefMathSciNetGoogle Scholar
  3. 3.
    Bachoc, C., Bannai, E., Coulangeon, R.: Codes and designs in Grassmannian spaces. Discrete Math. 277, 15–28 (2004) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bachoc, C., Coulangeon, R., Nebe, G.: Designs in Grassmannian spaces and lattices. J. Algebr. Comb. 16, 5–19 (2002) MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Böröczky Jr., K.: Finite Packing and Covering. Cambridge Tracts in Mathematics, vol. 154. Cambridge University Press, Cambridge (2004) MATHGoogle Scholar
  6. 6.
    Brouwer, A.E., Cohen, A.M., Neumaier, A.: Distance-Regular Graphs. Springer, Berlin (1989) MATHGoogle Scholar
  7. 7.
    Bump, D.: Lie Groups. Graduate Texts in Mathematics, vol. 225. Springer, New York (2004) MATHGoogle Scholar
  8. 8.
    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. Comb. 9, 129–140 (1999) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Conway, J.H., Hardin, R.H., Sloane, N.J.A.: Packing lines, planes, etc.: packings in Grassmannian spaces. Exp. Math. 5, 139–159 (1996) MATHMathSciNetGoogle Scholar
  10. 10.
    Conway, J.H., Sloane, N.J.A.: Sphere Packings, Lattices and Groups, 2nd edn. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 290. Springer, New York (1993) MATHGoogle Scholar
  11. 11.
    Delsarte, P.: An algebraic approach to the association schemes of coding theory. Philips Res. Rep. Suppl. (1973), vi+97 Google Scholar
  12. 12.
    Delsarte, P., Goethals, J.M., Seidel, J.J.: Bounds for systems of lines, and Jacobi polynomials. Philips Res. Rep. (1975), pp. 91–105 Google Scholar
  13. 13.
    Fulton, W., Harris, J.: Representation Theory. Springer, New York (1991) MATHGoogle Scholar
  14. 14.
    Godsil, C.D.: Polynomial spaces. In: Proceedings of the Oberwolfach Meeting “Kombinatorik”, vol. 73 (1986), pp. 71–88 (1989) Google Scholar
  15. 15.
    Godsil, C.D., Rötteler, M., Roy, A.: Mutually unbiased subspaces, in preparation Google Scholar
  16. 16.
    Godsil, C.D., Roy, A.: Mutually unbiased bases, equiangular lines, and spin models. Eur. J. Comb. 30, 246–262 (2009) MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Goodman, R., Wallach, N.R.: Representations and Invariants of the Classical Groups. Encyclopedia of Mathematics and its Applications, vol. 68. Cambridge University Press, Cambridge (1998) MATHGoogle Scholar
  18. 18.
    Helgason, S.: Groups and Geometric Analysis. Pure and Applied Mathematics, vol. 113. Academic Press, Orlando (1984) MATHGoogle Scholar
  19. 19.
    Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1990) MATHGoogle Scholar
  20. 20.
    James, A.T., Constantine, A.G.: Generalized Jacobi polynomials as spherical functions of the Grassmann manifold. Proc. Lond. Math. Soc. (3) 29, 174–192 (1974) MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Khatirinejad, M.: On Weyl-Heisenberg orbits of equiangular lines. J. Algebr. Comb. 28, 333–349 (2008) MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Levenshtein, V.: On designs in compact metric spaces and a universal bound on their size. Discrete Math. 192, 251–271 (1998) MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000) MATHGoogle Scholar
  24. 24.
    Renes, J., Blume-Kohout, R., Scott, A.J., Caves, C.M.: Symmetric informationally complete quantum measurements. J. Math. Phys. 45, 2171 (2004) MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Roy, A., Scott, A.J.: Weighted complex projective 2-designs from bases: optimal state determination by orthogonal measurements. J. Math. Phys. 48, 072110 (2007) CrossRefMathSciNetGoogle Scholar
  26. 26.
    Scott, A.J.: Tight informationally complete quantum measurements. J. Phys. A 39, 13507–13530 (2006) MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Sepanski, M.R.: Compact Lie Groups. Graduate Texts in Mathematics, vol. 235. Springer, New York (2007) MATHGoogle Scholar
  28. 28.
    Stanley, R.P.: Enumerative Combinatorics, vol. 2. Cambridge Studies in Advanced Mathematics, vol. 62. Cambridge University Press, Cambridge (1999) Google Scholar
  29. 29.
    Wong, Y.-c.: Differential geometry of Grassmann manifolds. Proc. Nat. Acad. Sci. USA 57, 589–594 (1967) MATHCrossRefGoogle Scholar
  30. 30.
    Zauner, G.: Quantendesigns. Ph.D. thesis, University of Vienna (1999) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Institute for Quantum Information Science & Department of Mathematics and StatisticsUniversity of CalgaryCalgaryCanada

Personalised recommendations