On the sphere decoding complexity of high-rate multigroup decodable STBCs in asymmetric MIMO systems

A space-time block code (STBC) is said to be multigroup decodable if the information symbols encoded by it can be partitioned into two or more groups such that each group of symbols can be maximum-likelihood (ML) decoded independently of the other symbol groups. In this paper, we show that the upper triangular matrix encountered during the sphere decoding of a linear dispersion STBC can be rank-deficient even when the rate of the code is less than the minimum of the number of transmit and receive antennas. We then show that all known families of high-rate (rate greater than 1) multigroup decodable codes have rank-deficient matrix even when the rate is less than the number of transmit and receive antennas, and this rank-deficiency problem arises only in asymmetric MIMO systems when the number of receive antennas is strictly less than the number of transmit antennas. Unlike the codes with full-rank matrix, the complexity of the sphere decoding-based ML decoder for STBCs with rank-deficient matrix is polynomial in the constellation size, and hence is high. We derive the ML sphere decoding complexity of most of the known high-rate multigroup decodable codes, and show that for each code, the complexity is a decreasing function of the number of receive antennas.

Wireless Network-Coded Bidirectional Relaying Using Latin Squares for M-PSK Modulation

The design of modulation schemes for the physical layer network-coded two-way relaying scenario is considered with a protocol which employs two phases: multiple access (MA) phase and broadcast (BC) phase. It was observed by Koike-Akino et al. that adaptively changing the network coding map used at the relay according to the channel conditions greatly reduces the impact of MA interference which occurs at the relay during the MA phase and all these network coding maps should satisfy a requirement called the exclusive law. We show that every network coding map that satisfies the exclusive law is representable by a Latin Square and conversely, that this relationship can be used to get the network coding maps satisfying the exclusive law. The channel fade states for which the minimum distance of the effective constellation at the relay become zero are referred to as the singular fade states. For M - PSK modulation (M any power of 2), it is shown that there are (M-2/4 - M/2 + 1) M singular fade states. Also, it is shown that the constraints which the network coding maps should satisfy so that the harmful effects of the singular fade states are removed, can be viewed equivalently as partially filled Latin Squares (PFLS). The problem of finding all the required maps is reduced to finding a small set of maps for M - PSK constellations (any power of 2), obtained by the completion of PFLS. Even though the completability of M x M PFLS using M symbols is an open problem, specific cases where such a completion is always possible are identified and explicit construction procedures are provided. Having obtained the network coding maps, the set of all possible channel realizations (the complex plane) is quantized into a finite number of regions, with a specific network coding map chosen in a particular region. It is shown that the complex plane can be partitioned into two regions: a region in which any network coding map which satisfies the exclusive law gives the same best performance and a region in which the choice of the network coding map affects the performance. The quantization thus obtained analytically, leads to the same as the one obtained using computer search for M = 4-PSK signal set by Koike-Akino et al., when specialized for Simulation results show that the proposed scheme performs better than the conventional exclusive-OR (XOR) network coding and in some cases outperforms the scheme proposed by Koike-Akino et al.

A transform approach to linear network coding for acyclic networks with delay

The algebraic formulation for linear network coding in acyclic networks with each link having an integer delay is well known. Based on this formulation, for a given set of connections over an arbitrary acyclic network with integer delay assumed for the links, the output symbols at the sink nodes at any given time instant is a Fq-linear combination of the input symbols across different generations, where Fq denotes the field over which the network operates. We use finite-field discrete Fourier transform (DFT) to convert the output symbols at the sink nodes at any given time instant into a Fq-linear combination of the input symbols generated during the same generation. We call this as transforming the acyclic network with delay into n-instantaneous networks (n is sufficiently large). We show that under certain conditions, there exists a network code satisfying sink demands in the usual (non-transform) approach if and only if there exists a network code satisfying sink demands in the transform approach. Furthermore, assuming time invariant local encoding kernels, we show that the transform method can be employed to achieve half the rate corresponding to the individual source-destination mincut (which are assumed to be equal to 1) for some classes of three-source three-destination multiple unicast network with delays using alignment strategies when the zero-interference condition is not satisfied.

Generalized distributive law for ML decoding of STBCs: further results

The problem of designing good Space-Time Block Codes (STBCs) with low maximum-likelihood (ML) decoding complexity has gathered much attention in the literature. All the known low ML decoding complexity techniques utilize the same approach of exploiting either the multigroup decodable or the fast-decodable (conditionally multigroup decodable) structure of a code. We refer to this well known technique of decoding STBCs as Conditional ML (CML) decoding. In [1], we introduced a framework to construct ML decoders for STBCs based on the Generalized Distributive Law (GDL) and the Factor-graph based Sum-Product Algorithm, and showed that for two specific families of STBCs, the Toepltiz codes and the Overlapped Alamouti Codes (OACs), the GDL based ML decoders have strictly less complexity than the CML decoders. In this paper, we introduce a `traceback' step to the GDL decoding algorithm of STBCs, which enables roughly 4 times reduction in the complexity of the GDL decoders proposed in [1]. Utilizing this complexity reduction from `traceback', we then show that for any STBC (not just the Toeplitz and Overlapped Alamouti Codes), the GDL decoding complexity is strictly less than the CML decoding complexity. For instance, for any STBC obtained from Cyclic Division Algebras that is not multigroup or conditionally multigroup decodable, the GDL decoder provides approximately 12 times reduction in complexity compared to the CML decoder. Similarly, for the Golden code, which is conditionally multigroup decodable, the GDL decoder is only about half as complex as the CML decoder.

A matroidal framework for network-error correcting codes

Matroidal networks were introduced by Dougherty et al. and have been well studied in the recent past. It was shown that a network has a scalar linear network coding solution if and only if it is matroidal associated with a representable matroid. The current work attempts to establish a connection between matroid theory and network-error correcting codes. In a similar vein to the theory connecting matroids and network coding, we abstract the essential aspects of network-error correcting codes to arrive at the definition of a matroidal error correcting network. An acyclic network (with arbitrary sink demands) is then shown to possess a scalar linear error correcting network code if and only if it is a matroidal error correcting network associated with a representable matroid. Therefore, constructing such network-error correcting codes implies the construction of certain representable matroids that satisfy some special conditions, and vice versa.

Fault-tolerant secret key generation

Mobile nodes observing correlated data communicate using an insecure bidirectional switch to generate a secret key, which must remain concealed from the switch. We are interested in fault-tolerant secret key rates, i.e., the rates of secret key generated even if a subset of nodes drop out before the completion of the communication protocol. We formulate a new notion of fault-tolerant secret key capacity, and present an upper bound on it. This upper bound is shown to be tight when the random variables corresponding to the observations of nodes are exchangeable. Further, it is shown that one round of interaction achieves the fault-tolerant secret key capacity in this case. The upper bound is also tight for the case of a pairwise independent network model consisting of a complete graph, and can be attained by a noninteractive protocol.

An asymptotically optimal push-pull method for multicasting over a random network

We consider multicast flow problems where either all of the nodes or only a subset of the nodes may be in session. Traffic from each node in the session has to be sent to every other node in the session. If the session does not consist of all the nodes, the remaining nodes act as relays. The nodes are connected by undirected edges whose capacities are independent and identically distributed random variables. We study the asymptotics of the capacity region (with network coding) in the limit of a large number of nodes, and show that the normalized sum rate converges to a constant almost surely. We then provide a decentralized push-pull algorithm that asymptotically achieves this normalized sum rate.

Regenerating codes for errors and erasures in distributed storage

Regenerating codes are a class of codes proposed for providing reliability of data and efficient repair of failed nodes in distributed storage systems. In this paper, we address the fundamental problem of handling errors and erasures at the nodes or links, during the data-reconstruction and node-repair operations. We provide explicit regenerating codes that are resilient to errors and erasures, and show that these codes are optimal with respect to storage and bandwidth requirements. As a special case, we also establish the capacity of a class of distributed storage systems in the presence of malicious adversaries. While our code constructions are based on previously constructed Product-Matrix codes, we also provide necessary and sufficient conditions for introducing resilience in any regenerating code.

Improved perfect space-time block codes

Perfect space-time block codes (STBCs) are based on four design criteria-full-rateness, nonvanishing determinant, cubic shaping, and uniform average transmitted energy per antenna per time slot. Cubic shaping and transmission at uniform average energy per antenna per time slot are important from the perspective of energy efficiency of STBCs. The shaping criterion demands that the generator matrix of the lattice from which each layer of the perfect STBC is carved be unitary. In this paper, it is shown that unitariness is not a necessary requirement for energy efficiency in the context of space-time coding with finite input constellations, and an alternative criterion is provided that enables one to obtain full-rate (rate of complex symbols per channel use for an transmit antenna system) STBCs with larger normalized minimum determinants than the perfect STBCs. Further, two such STBCs, one each for 4 and 6 transmit antennas, are presented and they are shown to have larger normalized minimum determinants than the comparable perfect STBCs which hitherto had the best-known normalized minimum determinants.

Capacity of a Gaussian MAC with energy harvesting transmit nodes

We consider a Gaussian multiple access channel (GMAC) where the users are sensor nodes powered by energy harvesters. The energy harvesters may have finite or infinite buffer to store the harvested energy. First, we find the capacity region of a GMAC powered by transmit nodes with an infinite energy buffer. Next, we consider a GMAC with the transmitting nodes equipped with a finite energy buffer. Initially we assume perfect knowledge of the buffer state information at both the encoders and the decoder. We provide an achievable region for this case. We also generalize the achievable region when only partial information about buffer state is available at both the encoders and the decoder.

A novel MCMC algorithm for near-optimal detection in large-scale uplink mulituser MIMO systems

In this paper, we propose a low-complexity algorithm based on Markov chain Monte Carlo (MCMC) technique for signal detection on the uplink in large scale multiuser multiple input multiple output (MIMO) systems with tens to hundreds of antennas at the base station (BS) and similar number of uplink users. The algorithm employs a randomized sampling method (which makes a probabilistic choice between Gibbs sampling and random sampling in each iteration) for detection. The proposed algorithm alleviates the stalling problem encountered at high SNRs in conventional MCMC algorithm and achieves near-optimal performance in large systems with M-QAM. A novel ingredient in the algorithm that is responsible for achieving near-optimal performance at low complexities is the joint use of a randomized MCMC (R-MCMC) strategy coupled with a multiple restart strategy with an efficient restart criterion. Near-optimal detection performance is demonstrated for large number of BS antennas and users (e.g., 64, 128, 256 BS antennas/users).

Fast-Decodable MIDO Codes With Large Coding Gain

In this paper, a new method is proposed to obtain full-diversity, rate-2 (rate of two complex symbols per channel use) space-time block codes (STBCs) that are full-rate for multiple input double output (MIDO) systems. Using this method, rate-2 STBCs for 4 x 2, 6 x 2, 8 x 2, and 12 x 2 systems are constructed and these STBCs are fast ML-decodable, have large coding gains, and STBC-schemes consisting of these STBCs have a non-vanishing determinant (NVD) so that they are DMT-optimal for their respective MIDO systems. It is also shown that the Srinath-Rajan code for the 4 x 2 system, which has the lowest ML-decoding complexity among known rate-2 STBCs for the 4x2 MIDO system with a large coding gain for 4-/16-QAM, has the same algebraic structure as the STBC constructed in this paper for the 4 x 2 system. This also settles in positive a previous conjecture that the STBC-scheme that is based on the Srinath-Rajan code has the NVD property and hence is DMT-optimal for the 4 x 2 system.

A Phase Transition for the Uniform Distribution in the Pattern Maximum Likelihood Problem

In this paper, we consider the setting of the pattern maximum likelihood (PML) problem studied by Orlitsky et al. We present a well-motivated heuristic algorithm for deciding the question of when the PML distribution of a given pattern is uniform. The algorithm is based on the concept of a ``uniform threshold''. This is a threshold at which the uniform distribution exhibits an interesting phase transition in the PML problem, going from being a local maximum to being a local minimum.

On 2-D Non-Adjacent-Error Channel Models

In this work, we consider two-dimensional (2-D) binary channels in which the 2-D error patterns are constrained so that errors cannot occur in adjacent horizontal or vertical positions. We consider probabilistic and combinatorial models for such channels. A probabilistic model is obtained from a 2-D random field defined by Roth, Siegel and Wolf (2001). Based on the conjectured ergodicity of this random field, we obtain an expression for the capacity of the 2-D non-adjacent-errors channel. We also derive an upper bound for the asymptotic coding rate in the combinatorial model.

Scheduling for Stable and Reliable Communication Over Multiaccess Channels and Degraded Broadcast Channels

We model communication of bursty sources: 1) over multiaccess channels, with either independent decoding or joint decoding and 2) over degraded broadcast channels, by a discrete-time multiclass processor sharing queue. We utilize error exponents to give a characterization of the processor sharing queue. We analyze the processor sharing queue model for the stable region of message arrival rates, and show the existence of scheduling policies for which the stability region converges to the information-theoretic capacity region in an appropriate limiting sense.