Combinatorial constructions of packings in Grassmannian spaces

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.

[1]  R. Paley.  On Orthogonal Matrices , 1933 .

[2]  R. H. Hardin,et al.  A Group-Theoretic Framework for the Construction of Packings in Grassmannian Spaces , 1999, math/0208002.

[3]  Jonathan Jedwab,et al.  Constructions of complex equiangular lines from mutually unbiased bases , 2014, Des. Codes Cryptogr..

[4]  Tomás Kocák,et al.  Families of optimal packings in real and complex Grassmannian spaces , 2017 .

[5]  Qing Xiang,et al.  Pseudo-Paley graphs and skew Hadamard difference sets from presemifields , 2007, Des. Codes Cryptogr..

[6]  N. J. A. Sloane,et al.  Packing Lines, Planes, etc.: Packings in Grassmannian Spaces , 1996, Exp. Math..

[7]  A. Whiteman.  A family of difference sets , 1962 .

[8]  Alexander Pott,et al.  Finite Geometry and Character Theory , 1995 .

[9]  Robert W. Heath,et al.  Constructing Packings in Grassmannian Manifolds via Alternating Projection , 2007, Exp. Math..

[10]  J. Seidel,et al.  BOUNDS FOR SYSTEMS OF LINES, AND JACOBI POLYNOMIALS , 1975 .

[11]  Akihiro Munemasa,et al.  Equiangular lines in Euclidean spaces , 2014, J. Comb. Theory, Ser. A.

[12]  D. M. Appleby.  Symmetric informationally complete–positive operator valued measures and the extended Clifford group , 2005 .

[13]  A. J. Scott,et al.  Symmetric informationally complete positive-operator-valued measures: A new computer study , 2010 .

[14]  N. Sloane,et al.  A Family of Optimal Packings in Grassmannian Manifolds , 1998, math/0208003.

[15]  Jonathan Jedwab,et al.  Large sets of complex and real equiangular lines , 2015, J. Comb. Theory, Ser. A.

[16]  Emma Lehmer,et al.  On Residue Difference Sets , 1953, Canadian Journal of Mathematics.

[17]  John I. Haas,et al.  Maximal orthoplectic fusion frames from mutually unbiased bases and block designs , 2016, 1607.04546.

[18]  A. Robert Calderbank,et al.  Grassmannian Packings From Operator Reed–Muller Codes , 2010, IEEE Transactions on Information Theory.

[19]  Yuan Xu,et al.  Cubature Formulas on Spheres , 2013 .

[20]  Michael J. Ganley,et al.  Direct Product Difference Sets , 1977, J. Comb. Theory, Ser. A.

[21]  Dominique de Caen,et al.  Large Equiangular Sets of Lines in Euclidean Space , 2000, Electron. J. Comb..

[22]  Robert L. McFarland,et al.  A Family of Difference Sets in Non-cyclic Groups , 1973, J. Comb. Theory A.

[23]  G. Zauner,et al.  QUANTUM DESIGNS: FOUNDATIONS OF A NONCOMMUTATIVE DESIGN THEORY , 2011 .

[24]  Aidan Roy,et al.  Bounds for codes and designs in complex subspaces , 2008, 0806.2317.

[25]  J. Singer.  A theorem in finite projective geometry and some applications to number theory , 1938 .