Density of Spherically-Embedded Stiefel and Grassmann Codes

The density of a code is the fraction of the coding space covered by packing balls centered around the codewords. This paper investigates the density of codes in the complex Stiefel and Grassmann manifolds equipped with the chordal distance. The choice of distance enables the treatment of the manifolds as subspaces of Euclidean hyperspheres. In this geometry, the densest packings are not necessarily equivalent to maximum-minimum-distance codes. Computing a code's density follows from computing: i) the normalized volume of a metric ball and ii) the kissing radius, the radius of the largest balls one can pack around the codewords without overlapping. First, the normalized volume of a metric ball is evaluated by asymptotic approximations. The volume of a small ball can be well-approximated by the volume of a locally-equivalent tangential ball. In order to properly normalize this approximation, the precise volumes of the manifolds induced by their spherical embedding are computed. For larger balls, a hyperspherical cap approximation is used, which is justified by a volume comparison theorem showing that the normalized volume of a ball in the Stiefel or Grassmann manifold is asymptotically equal to the normalized volume of a ball in its embedding sphere as the dimension grows to infinity. Then, bounds on the kissing radius are derived alongside corresponding bounds on the density. Unlike spherical codes or codes in flat spaces, the kissing radius of Grassmann or Stiefel codes cannot be exactly determined from its minimum distance. It is nonetheless possible to derive bounds on density as functions of the minimum distance. Stiefel and Grassmann codes have larger density than their image spherical codes when dimensions tend to infinity. Finally, the bounds on density lead to refinements of the standard Hamming bounds for Stiefel and Grassmann codes.Comment: Two-column version (24 pages, 6 figures, 4 tables). To appear in IEEE Transactions on Information Theor

Quantization Bounds on Grassmann Manifolds and Applications to MIMO Communications

This paper considers the quantization problem on the Grassmann manifold \mathcal{G}_{n,p}, the set of all p-dimensional planes (through the origin) in the n-dimensional Euclidean space. The chief result is a closed-form formula for the volume of a metric ball in the Grassmann manifold when the radius is sufficiently small. This volume formula holds for Grassmann manifolds with arbitrary dimension n and p, while previous results pertained only to p=1, or a fixed p with asymptotically large n. Based on this result, several quantization bounds are derived for sphere packing and rate distortion tradeoff. We establish asymptotically equivalent lower and upper bounds for the rate distortion tradeoff. Since the upper bound is derived by constructing random codes, this result implies that the random codes are asymptotically optimal. The above results are also extended to the more general case, in which \mathcal{G}_{n,q} is quantized through a code in \mathcal{G}_{n,p}, where p and q are not necessarily the same. Finally, we discuss some applications of the derived results to multi-antenna communication systems.Comment: 26 pages, 7 figures, submitted to IEEE Transactions on Information Theory in Aug, 200

Sphere packing bounds in the Grassmann and Stiefel manifolds

Applying the Riemann geometric machinery of volume estimates in terms of curvature, bounds for the minimal distance of packings/codes in the Grassmann and Stiefel manifolds will be derived and analyzed. In the context of space-time block codes this leads to a monotonically increasing minimal distance lower bound as a function of the block length. This advocates large block lengths for the code design.Comment: Replaced with final version, 11 page

Metric Entropy of Homogeneous Spaces

For a (compact) subset $K$ of a metric space and $\varepsilon > 0$, the {\em covering number} $N(K , \varepsilon )$ is defined as the smallest number of balls of radius $\varepsilon$ whose union covers $K$. Knowledge of the {\em metric entropy}, i.e., the asymptotic behaviour of covering numbers for (families of) metric spaces is important in many areas of mathematics (geometry, functional analysis, probability, coding theory, to name a few). In this paper we give asymptotically correct estimates for covering numbers for a large class of homogeneous spaces of unitary (or orthogonal) groups with respect to some natural metrics, most notably the one induced by the operator norm. This generalizes earlier author's results concerning covering numbers of Grassmann manifolds; the generalization is motivated by applications to noncommutative probability and operator algebras. In the process we give a characterization of geodesics in $U(n)$ (or $SO(m)$) for a class of non-Riemannian metric structures