Node Repair for Distributed Storage Systems over Fading Channels

Distributed storage systems and associated storage codes can efficiently store a large amount of data while ensuring that data is retrievable in case of node failure. The study of such systems, particularly the design of storage codes over finite fields, assumes that the physical channel through which the nodes communicate is error-free. This is not always the case, for example, in a wireless storage system. We study the probability that a subpacket is repaired incorrectly during node repair in a distributed storage system, in which the nodes communicate over an AWGN or Rayleigh fading channels. The asymptotic probability (as SNR increases) that a node is repaired incorrectly is shown to be completely determined by the repair locality of the DSS and the symbol error rate of the wireless channel. Lastly, we propose some design criteria for physical layer coding in this scenario, and use it to compute optimally rotated QAM constellations for use in wireless distributed storage systems.Comment: To appear in ISITA 201

Lattice Codes for Physical Layer Communications

Lattices are deceptively simple mathematical structures that have become indispensable for code design for physical layer communications. While lattice-related problems are interesting in their own right, the usefulness of these discrete structures in wireless communications provides additional motivation for their study and enables a multidisciplinary line of research.  This thesis is devoted to the study of lattice code design for physical layer communications. Modern wireless communication networks are required to accommodate significantly varied types of mobile devices, differing in available computational power or number of equipped antennas. Additionally, the density of the networks increases rapidly, and many communication protocols diverge from the classical direct point-to-point transmission in favor of allowing for intermediate relays to process and forward data. An important consequence of this shift towards more sophisticated transmission protocols is that traditional well-performing codes become futile for modern communications, thus the study and development of novel codes is called for.  Yet, however involved a transmission protocol may be, the characteristics of the physical medium, i.e., the wireless channel, stay unaffected. It is thus natural that an underlying lattice structure for code design remains crucial. This thesis consists of several articles considering lattice code design for four different communication settings relevant in modern wireless communications.  We begin by studying two communication scenarios for which space-time lattice codes, objects studied since 1998, arise naturally due to the characteristics of the transmission protocol. The first considered setup is an asymmetric point-to-point channel, for which we construct full-rank matrix lattices which serve as the underlying structure for code construction. In particular, we are interested in an invariant of certain orders in the considered algebra, called discriminant. We then move on to a relaying technique known as amplify-and-forward. We propose constructions of well-performing space-time lattice codes adopted to this particular setting, which additionally allow for a significant reduction in decoding complexity. The other two scenarios considered make use of Voronoi codes, also referred to as nested lattice codes, a concept first introduced in 1983. We study maximum-likelihood decoding in the context of a relaying protocol known as compute-and-forward, and are furthermore concerned with the message recoverability at the destination. Finally, we consider the wiretap channel and are particularly interested in the performance with respect to a potential eavesdropper. In the latter two scenarios, we derive design criteria related to the theta series of certain lattices involved in the code design

Fast-Decodable Space-Time Codes for the N-Relay and Multiple-Access MIMO Channel

In this article, the first general constructions of fast-decodable, more specifically (conditionally) g-group decodable, space-time block codes for the nonorthogonal amplify and forward (NAF) multiple-input multiple-output (MIMO) relay channel under the half-duplex constraint are proposed. In this scenario, the source and the intermediate relays used for data amplification are allowed to employ multiple antennas for data transmission and reception. The worst-case decoding complexity of the obtained codes is reduced by up to 75%. In addition to being fast-decodable, the proposed codes achieve full-diversity and have nonvanishing determinants, which has been shown to be useful for achieving the optimal diversity-multiplexing tradeoff (DMT) of the NAF channel. Furthermore, it is shown that the same techniques as in the cooperative scenario can be utilized to achieve fast-decodability for K-user MIMO multiple-access channel (MAC) space-time block codes. The resulting codes in addition exhibit the conditional nonvanishing determinant property which, for its part, has been shown to be useful for achieving the optimal MAC-DMT.Peer reviewe

Coded Caching Clusters with Device-to-Device Communications

We consider a geographically constrained caching community where popular data files are cached on mobile terminals and distributed through Device-to-Device (D2D) communications. To ensure availability, data files are protected against user mobility, or churn, with select caching and erasure coding methods. Communication and storage costs are considered, with an objective of minimizing the consumption of radio resources, given an available storage size. We focus on finding the coding method that minimizes the overall cost. Closed-form expressions for the expected consumption of radio resources incurred by data delivery and redundancy maintenance are derived. Closed form transmission costs in a circular caching community with a specific node density and caching method are calculated, when cost obeys a power law of distance. Our results are illustrated by numerical examples and verified by extensive computer simulations.Peer reviewe

An approximation of theta functions with applications to communications

Computing the theta series of an arbitrary lattice, and more specifically a related quantity known as the flatness factor, has been recently shown to be important for lattice code design in various wireless communication setups. However, the theta series is in general not known in closed form, excluding a small set of very special lattices. In this article, motivated by the practical applications as well as the mathematical problem itself, a simple approximation of the theta series of a lattice is derived. A rigorous analysis of its accuracy is provided. In relation to this, maximum-likelihood decoding in the context of compute-and-forward relaying is studied. Following previous work, it is shown that the related metric can exhibit a flat behavior, which can be characterized by the flatness factor of the decoding function. Contrary to common belief, we note that the decoding metric can be rewritten as a sum over a random lattice only when at most two sources are considered. Using a particular matrix decomposition, a link between the random lattice and the code lattice employed at the transmitter is established, which leads to an explicit criterion for code design, in contrast to implicit criteria derived previously. Finally, candidate lattices are examined with respect to the proposed criterion using the derived theta series approximation.Peer reviewe