Families of optimal packings in real and complex Grassmannian spaces
type: Item
instance of: scholarly article
title: Families of optimal packings in real and complex Grassmannian spaces
zbMATH Open document ID: 1381.52026
author: Tomáš Kocák
author: Martin Niepel
published in: Journal of Algebraic Combinatorics
publication date: 2017-02-16T00:00:00Z
review text: The problem of optimal packing in the real Grassmannian \(G_{\mathbb R}(m, n)\) of \(n\)-planes in \(\mathbb R^m\) as originally introduced in [\textit{J. H. Conway} et al., Exp. Math. 5, No. 2, 139--159 (1996; Zbl 0864.51012)] consists of finding a set of points \(U_1, U_2, \dots , U_N\) in the Grassmannian (i.e. \(n\)-dimensional subspaces of \(\mathbb R^m\)) such that the minimal distance between pairs of points, \(\min_{i\neq j} d (U_i, U_j)\), is maximal. The distance used is the chordal distance \(d (U, V)\) that can be expressed in terms of projection operators onto \(n\)-planes in \(\mathbb R^m\) as \(d^2(U, V) = \frac 1 2\mathrm{tr}\, (\mathcal P_U - \mathcal P_V)^2 = n - \mathrm{tr}\, (\mathcal P_U \mathcal P_V),\) where \(\mathcal P_U, \mathcal P_V\) are the matrices of the orthogonal projections on \(U\) and \(V.\) A projection matrix is closely related to the reflection matrix in the same plane, namely \(R_U = 2 \mathcal P_U - I\). The identification of an \(n\)-plane (a point in \(G_{\mathbb R}(m, n)\) or \(G_{\mathbb C}(m, n)\)) with the operator of orthogonal projection in \(\mathbb R^m\) or \(\mathbb C^m\) onto that plane defines an inclusion (actually, an isometric equivariant embedding) into the Euclidean space of real symmetric (resp. Hermitian) \(m \times m\) matrices equipped with the standard trace metric, the image belonging to a certain sphere centered at \(\frac{n}{m} I\). (See the above paper or the reviewer's [Publ. Inst. Math., Nouv. Sér. 59(73), 131--137 (1996; Zbl 0965.53038)]). Because of that, the classical simplex and orthoplex Rankin bounds for spherical packings can be translated to Grassmannian packings. In the space of \(m\times m\) real symmetric matrices, which has the dimension \(D = m(m+1)/2,\) the authors define an \(m \times m\) real (orthogonal) matrix orthoplex as a collection of \(2D\) orthogonal, real symmetric \(m\times m\) matrices \(R_1, R_2, \dots , R_{2D}\) satisfying \[ \mathrm{tr}\, (R_i^T R_j) = \begin{cases} m…
Mathematics Subject Classification ID: 52C17
Mathematics Subject Classification ID: 51F25
Mathematics Subject Classification ID: 51M20
Mathematics Subject Classification ID: 15A30
Mathematics Subject Classification ID: 15B34
Mathematics Subject Classification ID: 94B60
Mathematics Subject Classification ID: 14M15
zbMATH DE Number: 6685153
zbMATH Keywords: Grassmannian packings
zbMATH Keywords: optimal packings
zbMATH Keywords: Rankin bound
zbMATH Keywords: chordal distance
zbMATH Keywords: Hadamard matrices
zbMATH Keywords: space-time codes
reviewed by: Ivko Dimitrić
MaRDI profile type: Publication
full work available at URL: s10801-016-0702-x
OpenAlex ID: W2477460231
cites work: A group-theoretic framework for the construction of packings in Grassmannian spaces
cites work: Grassmannian Packings From Operator Reed–Muller Codes
cites work: Q3407104
cites work: On block coherence of frames
cites work: Q3411976
cites work: Packing Lines, Planes, etc.: Packings in Grassmannian Spaces
cites work: Constructing Packings in Grassmannian Manifolds via Alternating Projection
cites work: Q4146667
cites work: Round robin scheduling -- a survey
cites work: Bounds for codes and designs in complex subspaces
cites work: A family of optimal packings in Grassmannian manifolds
cites work: Grassmannian frames with applications to coding and communication
cites work: On the existence of equiangular tight frames
DOI: 10.1007/S10801-016-0702-X
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description: scientific article; zbMATH DE number 6685153