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

Nonintersecting Subspaces Based on Finite Alphabets

Two subspaces of a vector space are here called ``nonintersecting'' if they meet only in the zero vector. The following problem arises in the design of noncoherent multiple-antenna communications systems. How many pairwise nonintersecting M_t-dimensional subspaces of an m-dimensional vector space V over a field F can be found, if the generator matrices for the subspaces may contain only symbols from a given finite alphabet A subseteq F? The most important case is when F is the field of complex numbers C; then M_t is the number of antennas. If A = F = GF(q) it is shown that the number of nonintersecting subspaces is at most (q^m-1)/(q^{M_t}-1), and that this bound can be attained if and only if m is divisible by M_t. Furthermore these subspaces remain nonintersecting when ``lifted'' to the complex field. Thus the finite field case is essentially completely solved. In the case when F = C only the case M_t=2 is considered. It is shown that if A is a PSK-configuration, consisting of the 2^r complex roots of unity, the number of nonintersecting planes is at least 2^{r(m-2)} and at most 2^{r(m-1)-1} (the lower bound may in fact be the best that can be achieved).Comment: 14 page

A Noncoherent Space-Time Code from Quantum Error Correction

In this work, we develop a space-time block code for noncoherent communication using techniques from the field of quantum error correction. We decompose the multiple-input multiple-output (MIMO) channel into operators from quantum mechanics, and design a non-coherent space time code using the quantum stabilizer formalism. We derive an optimal decoder, and analyze the former through a quantum mechanical lens. We compare our approach to a comparable coherent approach and a noncoherent differential approach, achieving comparable or better performance.Comment: 6 pages, one figure, accepted at the 53rd annual Conference on Information Sciences and System

Time-frequency Grassmannian signalling for MIMO multi-channel-frequency-flat systems

In this paper, we consider the application of non-coherent Grassmannian signalling in practical multi-channel-frequency-flat multiple-input multiple-output (MIMO) wireless communication systems. In these systems, Grassmannian signalling, originally developed for single-channel block-fading systems, is not readily applicable. In particular, in such systems, the channel coefficients are constant across time and frequency, which implies that spectrally-efficient signalling ought to be jointly structured over these domains. To approach this goal, we develop a concatenation technique that yields a spectrally-efficient time-frequency Grassmannian signalling scheme, which enables the channel coherence bandwidth to be regarded as an additional coherence time. This scheme is shown to achieve the high signal-to-noise ratio non-coherent capacity of MIMO channels when the fading coefficients are constant over a time-frequency block. This scheme is also applicable in fast fading systems with coherence bandwidth exceeding that of one subchannel. The proposed scheme is independent of the symbol duration, i.e., the channel use duration, and is thus compatible with the transmit filter designs in current systems.The work of the first and second authors is supported, in part, by the Natural Sciences and Engineering Research Council of Canada (NSERC). This work is also supported, in part, by Huawei Canada Co., Ltd., in part, by the Ontario Ministry of Economic Development and Innovation's ORF-RE (Ontario Research Fund-Research Excellence) program, and, in part, by the Ministerio de Ciencia e Innovacion (project number TEC2011-27723-C02-02). The associate editor coordinating the review of this paper and approving it for publication was Z. Wang.Fouad, YMM.; Gohary, RH.; Cabrejas Peñuelas, J.; Yanikomeroglu, H.; Calabuig Soler, D.; Roger Varea, S.; Monserrat Del Río, JF. (2015). Time-frequency Grassmannian signalling for MIMO multi-channel-frequency-flat systems. IEEE Communications Letters. 19(3):475-478. https://doi.org/10.1109/LCOMM.2014.2386873S47547819