Fast-Decodable Asymmetric Space-Time Codes from Division Algebras

Multiple-input double-output (MIDO) codes are important in the near-future wireless communications, where the portable end-user device is physically small and will typically contain at most two receive antennas. Especially tempting is the 4 x 2 channel due to its immediate applicability in the digital video broadcasting (DVB). Such channels optimally employ rate-two space-time (ST) codes consisting of (4 x 4) matrices. Unfortunately, such codes are in general very complex to decode, hence setting forth a call for constructions with reduced complexity. Recently, some reduced complexity constructions have been proposed, but they have mainly been based on different ad hoc methods and have resulted in isolated examples rather than in a more general class of codes. In this paper, it will be shown that a family of division algebra based MIDO codes will always result in at least 37.5% worst-case complexity reduction, while maintaining full diversity and, for the first time, the non-vanishing determinant (NVD) property. The reduction follows from the fact that, similarly to the Alamouti code, the codes will be subsets of matrix rings of the Hamiltonian quaternions, hence allowing simplified decoding. At the moment, such reductions are among the best known for rate-two MIDO codes. Several explicit constructions are presented and shown to have excellent performance through computer simulations.Comment: 26 pages, 1 figure, submitted to IEEE Trans. Inf. Theory, October 201

Algebraic number theory and code design for Rayleigh fading channels

Algebraic number theory is having an increasing impact in code design for many different coding applications, such as single antenna fading channels and more recently, MIMO systems. Extended work has been done on single antenna fading channels, and algebraic lattice codes have been proven to be an effective tool. The general framework has been settled in the last ten years and many explicit code constructions based on algebraic number theory are now available. The aim of this work is to provide both an overview on algebraic lattice code designs for Rayleigh fading channels, as well as a tutorial introduction to algebraic number theory. The basic facts of this mathematical field will be illustrated by many examples and by the use of a computer algebra freeware in order to make it more accessible to a large audience

Probability Estimates for Fading and Wiretap Channels from Ideal Class Zeta Functions

In this paper, new probability estimates are derived for ideal lattice codes from totally real number fields using ideal class Dedekind zeta functions. In contrast to previous work on the subject, it is not assumed that the ideal in question is principal. In particular, it is shown that the corresponding inverse norm sum depends not only on the regulator and discriminant of the number field, but also on the values of the ideal class Dedekind zeta functions. Along the way, we derive an estimate of the number of elements in a given ideal with a certain algebraic norm within a finite hypercube. We provide several examples which measure the accuracy and predictive ability of our theorems.Comment: 24 pages. Extends our earlier arxiv submission arxiv.1303.347

Gitter und Anwendungen

The meeting focussed on lattices and their applications in mathematics and information technology. The research interests of the participants varied from engineering sciences, algebraic and analytic number theory, coding theory, algebraic geometry to name only a few

Numerical simulations of instabilities in general relativity

General relativity, one of the pillars of our understanding of the universe, has been a remarkably successful theory. It has stood the test of time for more than 100 years and has passed all experimental tests so far. Most recently, the LIGO collaboration made the first-ever direct detection of gravitational waves, confirming a long-standing prediction of general relativity. Despite this, several fundamental mathematical questions remain unanswered, many of which relate to the global existence and the stability of solutions to Einstein’s equations. This thesis presents our efforts to use numerical relativity to investigate some of these questions. We present a complete picture of the end points of black ring instabilities in five dimensions. Fat rings collapse to Myers-Perry black holes. For intermediate rings, we discover a previously unknown instability that stretches the ring without changing its thickness and causes it to collapse to a Myers-Perry black hole. Most importantly, however, we find that for very thin rings, the Gregory-Laflamme instability dominates and causes the ring to break. This provides the first concrete evidence that in higher dimensions, the weak cosmic censorship conjecture may be violated even in asymptotically flat spacetimes. For Myers-Perry black holes, we investigate instabilities in five and six dimensions. In six dimensions, we demonstrate that both axisymmetric and non-axisymmetric instabilities can cause the black hole to pinch off, and we study the approach to the naked singularity in detail. Another question that has attracted intense interest recently is the instability of anti-de Sitter space. In this thesis, we explore how breaking spherical symmetry in gravitational collapse in anti-de Sitter space affects black hole formation. These findings were made possible by our new open source general relativity code, GRChombo, whose adaptive mesh capabilities allow accurate simulations of phenomena in which new length scales are produced dynamically. In this thesis, we describe GRChombo in detail, and analyse its performance on the latest supercomputers. Furthermore, we outline numerical advances that were necessary for simulating higher dimensional black holes stably and efficiently.My PhD was funded by an STFC studentship initially and by the European Research Council Grant No. ERC-2014-StG 639022-NewNGR in my final year. Furthermore, I received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant agreement No. 690904. The simulations presented in this thesis were carried out on the following supercomputers: *) The COSMOS Shared Memory system at DAMTP, University of Cambridge, operated on behalf of the STFC DiRAC HPC Facility. This sytem is funded by BIS National E-infrastructure capital Grant No.~ST/ J005673/1 and STFC Grants No.~ST/H008586/1, No.~ST/K00333X/1. *) MareNostrum III and MareNostrum IV at the Barcelona Supercomputing Centre through the grants FI-2016-3-0006 and PRACE Tier-0 PPFPWG respectively. *) Stampede and Stampede2 at the Texas Advanced Computing Center, University of Texas at Austin, through the NSF-XSEDE grant No.~PHY-090003 and an allocation provided by Intel for their Parallel Computing Centres. *) SuperMike-II at Louisiana State University under allocation NUMREL06. *) Cartesius, SURFsara, in the Netherlands through the PRACE DECI grant NRBA

High Spectral Efficiency Fiber-Optic Transmission Systems Using Pilot Tones

Modern fiber-optic communication systems combine state-of-the-art components with powerful digital signal processing (DSP) to maximize the system spectral efficiency (SE). Systems rely on wavelength-division multiplexing, including superchannel transmission, to enable transmission over the available bandwidth which reaches about 10 THz when accounting for the so-called C and L bands. A superchannel is a set of densely packed wavelength channels viewed as a single unit. By treating the channels together, they can be packed more closely than what is normally feasible and sharing of resources among the channels within the superchannel can be considered. In this thesis we focus on the special case of superchannels formed using coherent optical frequency combs. A frequency comb is a multi-wavelength light source and comb-based superchannels consists of channels which are modulated on lines originating from a common comb. Frequency combs have phase-locked carriers, meaning that in contrast to the standard case of independent lasers, the channels within a comb-based superchannel are locked on a frequency grid. Moreover, it implies that the carrier offsets originating from a non-ideal laser source are shared among all comb lines.Shared carrier offsets can be exploited to reduce the complexity of the DSP used to effectively recover the data. A frequency comb is fully characterized by knowing the state of two of its lines, meaning that if this information is transferred to the receiver, one could compensate carrier offsets for all wavelength channels within the superchannel. By transmission of optical pilot tones, self-homodyne detection of a 50x20Gbaud PM-64QAM superchannel is demonstrated with 4% spectral overhead. While two tones are required to fully phase-lock two combs, a single tone is enough to enable significant relaxation of the DSP-requirements while at the same time requiring minimal additional complexity compared to standard intradyne systems. Superchannel transmission using a single shared pilot tone is demonstrated by transmission of a 51x24Gbaud PM-128QAM superchannel with a resulting SE of 10.3bits/s/Hz. The single pilot scheme is also evaluated for distances up to 1000km showing high robustness to both noise and fiber nonlinearities. Finally, the high gain low overhead combination of the single pilot-tone scheme was used in a record demonstration reaching a SE of 11.5bits/s/Hz for fully loaded C-band transmission

Institute for Computational Mechanics in Propulsion (ICOMP) fourth annual review, 1989

The Institute for Computational Mechanics in Propulsion (ICOMP) is operated jointly by Case Western Reserve University and the NASA Lewis Research Center. The purpose of ICOMP is to develop techniques to improve problem solving capabilities in all aspects of computational mechanics related to propulsion. The activities at ICOMP during 1989 are described