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

SMARAD - Centre of Excellence in Smart Radios and Wireless Research - Activity Report 2011 - 2013

Centre of Excellence in Smart Radios and Wireless Research (SMARAD), originally established with the name Smart and Novel Radios Research Unit, is aiming at world-class research and education in Future radio and antenna systems, Cognitive radio, Millimetre wave and THz techniques, Sensors, and Materials and energy, using its expertise in RF, microwave and millimeter wave engineering, in integrated circuit design for multi-standard radios as well as in wireless communications. SMARAD has the Centre of Excellence in Research status from the Academy of Finland since 2002 (2002-2007 and 2008-2013). Currently SMARAD consists of five research groups from three departments, namely the Department of Radio Science and Engineering, Department of Micro and Nanosciences, and Department of Signal Processing and Acoustics, all within the Aalto University School of Electrical Engineering. The total number of employees within the research unit is about 100 including 8 professors, about 30 senior scientists and about 40 graduate students and several undergraduate students working on their Master thesis. The relevance of SMARAD to the Finnish society is very high considering the high national income from exports of telecommunications and electronics products. The unit conducts basic research but at the same time maintains close co-operation with industry. Novel ideas are applied in design of new communication circuits and platforms, transmission techniques and antenna structures. SMARAD has a well-established network of co-operating partners in industry, research institutes and academia worldwide. It coordinates a few EU projects. The funding sources of SMARAD are diverse including the Academy of Finland, EU, ESA, Tekes, and Finnish and foreign telecommunications and semiconductor industry. As a by-product of this research SMARAD provides highest-level education and supervision to graduate students in the areas of radio engineering, circuit design and communications through Aalto University and Finnish graduate schools. During years 2011 – 2013, 18 doctor degrees were awarded to the students of SMARAD. In the same period, the SMARAD researchers published 197 refereed journal articles and 360 conference papers