Combinatorial constructions of packings in Grassmannian spaces
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The problem of packing n-dimensional subspaces of m-dimensional Euclidean space such that these subspaces are as far apart as possible was introduced by Conway, Hardin and Sloane. It can be seen as a higher dimensional version of spherical codes or equiangular lines. In this paper, we first give a general construction of equiangular lines, and then present a family of equiangular lines with large size from direct product difference sets. Meanwhile, for packing higher dimensional subspaces, we give three constructions of optimal packings in Grassmannian spaces based on difference sets and Latin squares. As a consequence, we obtain many new classes of optimal Grassmannian packings.
KeywordsGrassmannian packing Equiangular line Difference set Latin square
Mathematics Subject ClassificationPrimary: 52C17 Secondary: 14M15 94B60
The authors express their gratitude to the anonymous reviewers for their detailed and constructive comments which are very helpful to the improvement of the presentation of this paper. Research supported by the National Natural Science Foundation of China under Grant Nos. 11431003 and 61571310, Beijing Hundreds of Leading Talents Training Project of Science and Technology, and Beijing Municipal Natural Science Foundation.
- 1.Appleby D.M.: Symmetric informationally complete-positive operator valued measures and the extended Clifford group. J. Math. Phys. 46(5), 052107, 29 (2005).Google Scholar
- 3.Bodmann B.G., Haas J.I.: Maximal orthoplectic fusion frames from mutually unbiased bases and block designs. arXiv: 1607.04546.
- 5.Colbourn C.J., Dinitz J.H. (eds.): Handbook of Combinatorial Designs. Discrete Mathematics and Its Applications (Boca Raton), 2nd edn. Chapman & Hall/CRC, Boca Raton (2007).Google Scholar
- 7.Davis J.A., Jedwab J.: A survey of Hadamard difference sets. In: Groups, Difference Sets, and the Monster (Columbus, OH, 1993), Ohio State University Mathematical Research Institute Publications, vol. 4, pp. 145–156. de Gruyter, Berlin (1996).Google Scholar
- 8.de Caen D.: Large equiangular sets of lines in Euclidean space. Electron. J. Comb. 7, R55 (2000).Google Scholar
- 17.König H.: Cubature formulas on spheres. In: Hauffmann W., Jetter K., Reimer M. (eds.) Advances in multivariate approximation (Witten-Bommerholz, 1998). Mathematical Research, vol. 107, pp. 201–211. Wiley, Berlin (1999).Google Scholar
- 22.Pott A.: Finite Geometry and Character Theory. Lecture Notes in Mathematics, vol. 1601. Springer, Berlin (1995).Google Scholar
- 24.Scott, A.J., Grassl, M.: Symmetric informationally complete positive-operator-valued measures: a new computer study. J. Math. Phys. 51(4), 042203, 16 (2010).Google Scholar