Families of optimal packings in real and complex Grassmannian spaces
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 \[ \mathrm{tr}\, (R_i^T R_j) = \begin{cases} m \quad &\text{if }i\neq j,\\ -m \quad &\text{if }\{i, j \} = \{2k-1, 2k \},\\ 0 \quad &\text{otherwise.}\end{cases} \] Similarly, an \(m\times m\) complex (unitary) matrix orthoplex is defined. In [Conway et al., loc. cit.], several optimal packings are determined, including examples of optimal orthoplices of planes in \(\mathbb R^4\) and 4-dimensional subspaces in \(\mathbb R^8\). In [\textit{P. W. Shor} and \textit{N. J. A. Sloane}, J. Algebr. Comb. 7, No. 2, 157--163 (1998; Zbl 0904.52009)], certain optimal collections of \(2^{2k} + 2^k - 2\) real subspaces of dimension \(2^{k-1}\) in \(\mathbb R^{2^k}\) were produced. The investigation of Grassmannian packings was subsequently extended to the complex Grassmannian \(G_{\mathbb C}(m, n)\) and the existence of the optimal configurations of \(2^{2k+1} - 2\) complex subspaces of dimension \(2^{k-1}\) in \(\mathbb C^{2^k}\) was established in [\textit{A. Ashikhmin} and \textit{A. R. Calderbank}, ``Space-time Reed-Muller codes for noncoherent MIMO transmission'', in: Proceedings of the international symposium on information theory, ISIT 2005, 4--9 September 2005. SA: Adelaide. 1952--1956 (2005; \url{doi:10.1109/ISIT.2005.1523686})]. All of these results combined settled the question of existence of optimal orthoplex Grassmannian packings for real and complex Grassmannians of subspaces of mid-dimension in real resp. complex Euclidean space whose dimension is a power of 2. In the paper under review, the authors produce, using Hadamard matrices, a new family of optimal orthoplex packings in \(G_{\mathbb R}(8l, 4l)\) and \(G_{\mathbb C}(4l, 2l)\) and a related optimal simplex packings in \(G_{\mathbb R}(8l-1, 4l-1)\) and \(G_{\mathbb R}(8l-1, 4l)\). More precisely, they prove the following results: If there exists a \(4l\times 4l\) Hadamard matrix (\(l \in \mathbb N\)), then there exists an optimal orthoplex packing of \((8l)^2 + 8l - 2\) real subspaces of dimension \(4l\) in the real Grassmannian \(G_{\mathbb R}(8l, 4l)\). Under the same assumption, there exists an optimal orthoplex packing of \(2(4l)^2 - 2\) complex subspaces of dimension \(2l\) in \(G_{\mathbb C}(4l, 2l)\). Moreover, if there exists a \(4l\times 4l\) Hadamard matrix, an \(8l\times 8l\) skew Hadamard matrix and a related 1-factorization of a complete graph \(K_{8l}\), then there exists an optimal simplex packing of \(8l (8l-1)/2\) real subspaces of dimension \(4l-1\) in \(G_{\mathbb R}(8l-1, 4l-1)\), whereas their orthogonal complements form an optimal simplex packing in \(G_{\mathbb R}(8l-1, 4l)\). We recall that an \(n\times n\) Hadamard matrix \(H\) is a matrix with entries equal to \(\pm 1\), satisfying \(H^TH = n I\) and that a skew Hadamard matrix satisfies additionally \(H^T + H = 2 I\). The paper concludes with a construction of a maximal optimal simplex packings in \(G_{\mathbb C}(2l - 1, l - 1)\) and \(G_{\mathbb C}(2l - 1, l)\).
scientific article; zbMATH DE number 6685153
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