Hsiao-feng Francis Lu

Maximal Orders in the Design of Dense Space-Time Lattice Codes

In this paper, we construct explicit rate-one, full-diversity, geometrically dense matrix lattices with large, nonvanishing determinants (NVDs) for four transmit antenna multiple-input-single-output (MISO) space-time (ST) applications. The constructions are based on the theory of rings of algebraic integers and related subrings of the Hamiltonian quaternions and can be extended to a larger number of Tx antennas. The usage of ideals guarantees an NVD larger than one and an easy way to present the exact proofs for the minimum determinants. The idea of finding denser sublattices within a given division algebra is then generalized to a multiple-input-multiple-output (MIMO) case with an arbitrary number of Tx antennas by using the theory of cyclic division algebras (CDAs) and maximal orders. It is also shown that the explicit constructions in this paper all have a simple decoding method based on sphere decoding. Related to the decoding complexity, the notion of sensitivity is introduced, and experimental evidence indicating a connection between sensitivity, decoding complexity, and performance is provided. Simulations in a quasi-static Rayleigh fading channel show that our dense quaternionic constructions outperform both the earlier rectangular lattices and the rotated quasi-orthogonal ABBA lattice as well as the diagonal algebraic space-time (DAST) lattice. We also show that our quaternionic lattice is better than the DAST lattice in terms of the diversity-multiplexing gain tradeoff (DMT).

Inverse Determinant Sums and Connections Between Fading Channel Information Theory and Algebra

This work considers inverse determinant sums, which arise from the union bound on the error probability, as a tool for designing and analyzing algebraic space-time block codes. A general framework to study these sums is established, and the connection between asymptotic growth of inverse determinant sums and the diversity-multiplexing gain tradeoff is investigated. It is proven that the growth of the inverse determinant sum of a division algebra-based space-time code is completely determined by the growth of the unit group. This reduces the inverse determinant sum analysis to studying certain asymptotic integrals in Lie groups. Using recent methods from ergodic theory, a complete classification of the inverse determinant sums of the most well-known algebraic space-time codes is provided. The approach reveals an interesting and tight relation between diversity-multiplexing gain tradeoff and point counting in Lie groups.

Diversity-multiplexing gain tradeoff: A tool in algebra?

Since the invention of space-time coding numerous algebraic methods have been applied in code design. In particular algebraic number theory and central simple algebras have been on the forefront of the research. In this paper we are turning the table and asking whether information theory can be used as a tool in algebra. We first show how diversity-multiplexing gain tradeoff (DMT) bounds of Zheng and Tse will give us information of the spread of determinants in matrix lattices and then apply these results to analyze unit groups of orders of division algebras. The results considering unit groups are not new or the best possible but we do find that this interesting relation between algebra and information theory is quite surprising and worth pointing out.

Space-time storage codes for wireless distributed storage systems

Distributed storage systems (DSSs) have gained a lot of interest recently, thanks to their robustness and scalability compared to single-device storage. Majority of the related research has exclusively concerned the network layer. At the same time, the number of users of, e.g., peer-to-peer (p2p) and device-to-device (d2d) networks as well as proximity based services is growing rapidly, and the mobility of users is considered more and more important. This motivates, in contrast to the existing literature, the study of the physical layer functionality of wireless distributed storage systems. In this paper, we take the first step towards protecting the storage repair transmissions from physical layer errors when the transmission takes place over a fading channel. To this end, we introduce the notion of a space-time storage code, drawing together the aspects of network layer and physical layer functionality and resulting in cross-layer robustness. It is also pointed out that existing space-time codes are too complex to be utilized in storage networks when the number of helpers involved is larger than the number of receive antennas at the newcomer or data collector, hence creating a call for less complex transmission protocols.

Remarks on Diversity-Multiplexing Tradeoffs for Multiple-Access and Point-to-Point MIMO Channels

In this paper, we answer several open questions related to diversity-multiplexing tradeoffs (DMTs) for point-to-point and multiple-access (MAC) MIMO channels. By analyzing the DMT performance of a simple code, we show that the optimal MAC-DMT holds even when the channel remains fixed for less than Knt+nr-1 channel uses, where K is the number of users, nt is the number of transmit antennas of each user, and nr is the number of receive antennas at receiver. We also prove that the simple code is MAC-DMT optimal. A general code design criterion for constructing MAC-DMT optimal codes that is much more relaxed than the previously known design criterion is provided. Finally, by changing some design parameters, the simple code is modified for use in point-to-point MIMO channels. We show the modified code achieves the same DMT performance as the Gaussian random code.

Efficiently sphere-decodable physical layer transmission schemes for wireless storage networks

Three transmission schemes over a new type of multiple-access channel (MAC) model with inter-source communication links are proposed and investigated in this paper. This new channel model is well motivated by, e.g., wireless distributed storage networks, where communication to repair a lost node takes place from helper nodes to a repairing node over a wireless channel. Since in many wireless networks nodes can come and go in an arbitrary manner, there must be an inherent capability of inter-node communication between every pair of nodes.Assuming that communication is possible between every pair of helper nodes, the newly proposed schemes are based on various smart time-sharing and relaying strategies. In other words, certain helper nodes will be regarded as relays, thereby converting the conventional uncooperative multiple-access channel to a multiple-access relay channel (MARC). The diversity-multiplexing gain tradeoff (DMT) of the system together with efficient sphere-decodability and low structural complexity in terms of the number of antennas required at each end is used as the main design objectives. While the optimal DMT for the new channel model is fully open, it is shown that the proposed schemes outperform the DMT of the simple time-sharing protocol and, in some cases, even the optimal uncooperative MAC DMT.While using a wireless distributed storage network as a motivating example throughout the paper, the MAC transmission techniques proposed here are completely general and as such applicable to any MAC communication with inter-source communication links.

Coding schemes with constant sphere-decoding complexity and high DMT performance for MIMO multiple access channels with low-rate feedback

In a MIMO MAC where the base station has fewer receive antennas than the transmit antennas of all users, sphere (lattice) decoding for the existing MIMO-MAC codes requires an exhaustive search of exponentially large size before processing the root of a sphere-decoding tree. In this paper, two coding schemes are proposed and are shown to yield a constant sphere-decoding complexity, independent of the numbers of users and transmit antennas. The schemes require a channel feedback, but only at an extremely low rate. The first scheme is based on user selection, and the second scheme selects jointly users and transmit antennas, using a fast antenna selection algorithm recently proposed by Jiang and Varanasi. It also involves a design of rate-assignments that maximizes the overall DMT performance. It is shown that both schemes yield DMT performances far superior to the optimal MIMO-MAC DMT without channel feedback in certain multiplexing gain regime. Simulation results confirm that in some cases the second proposed scheme can provide an astonishing SNR gain of 14:5 dB at outage probability 10-6 compared to the optimal coding schemes without feedback.

Optimal Distributed Codes for Feedback-Aided Cooperative Relay Networks

A novel transmission scheme for cooperative relay networks is presented in this paper. The proposed scheme is based on the non-orthogonal selection decode-and-forward protocol with an additional assumption of having a low rate feedback channel from the destination to relays. Benefited from the feedback information, an optimal distributed code that has an extremely short delay equal to four is constructed, and the same code is applicable to networks with the arbitrary number of relays to yield optimal cooperative diversity. The proposed code is sphere decodable with a decoding complexity again independent of the number of relays in high SNR regime. In particular, when operating at multiplexing gain ≥(1/2), the lattice decoder at the destination has a zero complexity exponent, meaning a constant decoding complexity and independent of transmission rate. Analyses for the decoding complexity of other existing diversity-optimal distributed codes are also provided. It is shown that these codes have a linear growth in delay and an exponential growth in decoding complexity as the number of relays increases.

An Error Event Sensitive Tradeoff Between Rate and Coding Gain in MIMO MAC

This paper investigates the design of codes for multiple-input multiple-output (MIMO) multiple access channel (MAC). If a joint maximum-likelihood decoding is to be performed at the receiver, then every MIMO-MAC code can be regarded as a single-user code, where the minimum determinant criterion proposed by Tarokh et al. is useful for designing such codes and for upper bounding the maximum pairwise error probability (PEP), whenever the codes are of finite rate and operate in finite signal-to-noise ratio range. Unlike the case of single-user codes where the minimum determinant can be lower bounded by a fixed constant as code-rate grows, it was proved by Lahtonen et al. that the minimum determinant of MIMO-MAC codes decays as a function of the rates. This decay phenomenon is further investigated in this paper, and upper bounds for the decays of minimum determinant corresponding to each error event are provided. Lower bounds for the optimal decay are established and are based on an explicit construction of codes using algebraic number theory and Diophantine approximation. For some error profiles, the constructed codes are shown to meet the aforementioned upper bounds, hence they are optimal finite-rate codes in terms of PEPs associated with such error events. An asymptotic diversity-multiplexing gain tradeoff (DMT) analysis of the proposed codes is also given. It is shown that these codes are DMT optimal when the values of multiplexing gains are small.

Analysis and practice of uniquely decodable one-to-one code

In this paper, we consider the uniquely decodable one-to-one code (UDOOC) that is obtained by inserting a comma indicator, termed the unique word (UW), between consecutive one-to-one codewords for separation. As such, we analyze a class of UDOOCs and present practical algorithms for encoding and decoding such codes. Specifically, for various cases of UWs, we investigate the number of length-n codewords of UDOOCs and their asymptotic growth rates in n. The proposed encoding and decoding algorithms of UDOOCs can be implemented in parallel at low computational complexity without storing the codebook. Simulation results show that for proper choices of UWs, UDOOCs can achieve better compression efficiency than Lempel-Ziv codes even when the source is not statistically independent.