Thomas A. Courtade

Soft Information for LDPC Decoding in Flash: Mutual-Information Optimized Quantization

High-capacity NAND flash memory can achieve high density storage by using multi-level cells (MLC) to store more than one bit per cell. Although this larger storage capacity is certainly beneficial, the increased density also increases the raw bit error rate (BER), making powerful error correction coding necessary. Traditional flash memories employ simple algebraic codes, such as BCH codes, that can correct a fixed, specified number of errors. This paper investigates the application of low-density parity-check (LDPC) codes which are well known for their ability to approach capacity in the AWGN channel. We obtain soft information for the LDPC decoder by performing multiple cell reads with distinct word-line voltages. The values of the word-line voltages (also called reference voltages) are optimized by maximizing the mutual information between the input and output of the multiple-read channel. Our results show that using this soft information in the LDPC decoder provides a significant benefit and enables us to outperform BCH codes over a range of block error rates.

Multiterminal Source Coding Under Logarithmic Loss

We consider the classical two-encoder multiterminal source coding problem where distortion is measured under logarithmic loss. We provide a single-letter description of the achievable rate distortion region for all discrete memoryless sources with finite alphabets. By doing so, we also give the rate distortion region for the m-encoder CEO problem (also under logarithmic loss). Several applications and examples are given.

Coded Cooperative Data Exchange in Multihop Networks

Consider a connected network of n nodes that all wish to recover k desired packets. Each node begins with a subset of the desired packets and exchanges coded packets with its neighbors. This paper provides necessary and sufficient conditions that characterize the set of all transmission strategies that permit every node to ultimately learn (recover) all k packets. When the network satisfies certain regularity conditions and packets are randomly distributed, this paper provides tight concentration results on the number of transmissions required to achieve universal recovery. For the case of a fully connected network, a polynomial-time algorithm for computing an optimal transmission strategy is derived. An application to secrecy generation is discussed.

Enhanced Precision Through Multiple Reads for LDPC Decoding in Flash Memories

Multiple reads of the same Flash memory cell with distinct word-line voltages provide enhanced precision for LDPC decoding. In this paper, the word-line voltages are optimized by maximizing the mutual information (MI) of the quantized channel. The enhanced precision from a few additional reads allows frame error rate (FER) performance to approach that of full-precision soft information and enables an LDPC code to significantly outperform a BCH code. A constant-ratio constraint provides a significant simplification in the optimization with no noticeable loss in performance. For a well-designed LDPC code, the quantization that maximizes the mutual information also minimizes the FER in our simulations. However, for an example LDPC code with a high error floor caused by small absorbing sets, the MMI quantization does not provide the lowest frame error rate. The best quantization in this case introduces more erasures than would be optimal for the channel MI in order to mitigate the absorbing sets of the poorly designed code. The paper also identifies a trade-off in LDPC code design when decoding is performed with multiple precision levels; the best code at one level of precision will typically not be the best code at a different level of precision.

Optimal exchange of packets for universal recovery in broadcast networks

Consider an arbitrarily connected broadcast network of N nodes that all wish to recover k desired packets. Each node begins with a subset of the desired packets and broadcasts messages to its neighbors. For the case where nodes must transmit an integer number of packets, this paper provides necessary and sufficient conditions which characterize the set of all transmission schemes that permit universal recovery (in which every node learns all k packets). By relaxing the integer transmission constraint, this paper gives a computable lower-bound on the amount of information required to be broadcast to achieve universal recovery. Furthermore, a network-coding-based scheme (computable in polynomial time) can always achieve this lower bound if packet splitting is permitted. In this way, packet splitting can provide a significant reduction in the amount of communication required for universal recovery. For cliques with N nodes, this paper shows that splitting the packet into N - 1 chunks allows the lower bound to be achieved with high probability.

Optimal Allocation of Redundancy Between Packet-Level Erasure Coding and Physical-Layer Channel Coding in Fading Channels

For a block-fading channel, this paper optimizes the allocation of redundancy between packet-level erasure coding (which provides additional packets to compensate for packet loss) and physical layer channel coding (which lowers the probability of packet loss). After some manipulation, standard optimization techniques determine the trade-off between the amount of packet-level erasure coding and physical-layer channel coding that minimizes the transmit power required to provide reliable communication. Our results indicate that the optimal combination of packet-level erasure coding and physical-layer coding provides a significant benefit over pure physical-layer coding when no form of channel diversity is present within a packet transmission. However, the benefit of including packet-level erasure coding diminishes as more diversity becomes available within a packet transmission. Even with no diversity within a packet transmission, this paper shows that as the total redundancy becomes large the optimal redundancy for packet-level erasure coding reaches a limit while the optimal redundancy for physical-layer coding continues to increase. Hence providing limitless redundancy at the packet-level with rateless codes such as fountain codes is not the best use of limitless redundancy for block-fading channels.

Weighted universal recovery, practical secrecy, and an efficient algorithm for solving both

In this paper, we consider a network of n nodes, each initially possessing a subset of packets. Each node is permitted to broadcast functions of its own packets and the messages it receives to all other nodes via an error-free channel. We provide an algorithm that efficiently solves the Weighted Universal Recovery Problem and the Secrecy Generation Problem for this network. In the Weighted Universal Recovery Problem, the goal is to design a sequence of transmissions that ultimately permits all nodes to recover all packets initially present in the network. We show how to compute a transmission scheme that is optimal in the sense that the weighted sum of the number of transmissions is minimized. For the Secrecy Generation Problem, the goal is to generate a secret-key among the nodes that cannot be derived by an eavesdropper privy to the transmissions. In particular, we wish to generate a secret-key of maximum size. Further, we discuss private-key generation, which applies to the case where a subset of nodes is compromised by the eavesdropper. For the network under consideration, both of these problems are combinatorial in nature. We demonstrate that each of these problems can be solved efficiently and exactly. Notably, we do not require any terms to grow asymptotically large to obtain our results. This is in sharp contrast to classical information-theoretic problems despite the fact that our problems are information-theoretic in nature. Finally, the algorithm we describe efficiently solves an Integer Linear Program of a particular form. Due to the general form we consider, it may prove useful beyond these applications.

A Cross-Layer Perspective on Rateless Coding for Wireless Channels

Rateless coding ensures reliability by providing ever-increasing redundancy, traditionally at the packet level (i.e. the application layer) through erasure coding. This paper explores whether additional redundancy for wireless channels is most helpful at the packet level through erasure coding or at the physical layer through lower-rate channel coding. This cross-layer trade-off is explored in a traditional wireless setting where the communication of a message consisting of a fixed number of packets takes place over a Rayleigh fading channel. The examined scenarios include both a single receiver and multiple cooperating receivers allowing the results to be extended to situations where selection diversity is available in the system. For several interesting scenarios, this paper determines the optimal trade-off between the amount of packet-level erasure coding and physical-layer channel coding required to provide reliable communication over the widest range of operating SNRs. Our results indicate that packet-level erasure coding can provide a significant benefit when no other form of diversity is available. In many cases, the amount of redundancy that should be allocated to such erasure coding is nearly constant, and further redundancy (i.e. any rateless coding) should be applied to the physical layer.

Design of Energy- and Cost-Efficient Massive MIMO Arrays

Large arrays of radios have been exploited for beamforming and null steering in both radar and communication applications, but cost and form factor limitations have precluded their use in commercial systems. This paper discusses how to build arrays that enable multiuser massive multiple-input-multiple-output (MIMO) and aggressive spatial multiplexing with many users sharing the same spectrum. The focus of the paper is the energy- and cost-efficient realization of these arrays in order to enable new applications. Distributed algorithms for beamforming are proposed, and the optimum array size is considered as a function of the performance of the receiver, transmitter, frequency synthesizer, and signal distribution within the array. The effects of errors such as phase noise and synchronization skew across the array are analyzed. The paper discusses both RF frequencies below 10 GHz, where fully digital techniques are preferred, and operation at millimeter (mm)-wave bands where a combination of digital and analog techniques are needed to keep cost and power low.

Efficient universal recovery in broadcast networks

Consider a connected broadcast network of N nodes that all wish to recover k desired packets. Each node begins with a subset of the desired packets and broadcasts messages to its neighbors. In a previous paper we established necessary and sufficient conditions on the number of transmissions from each node required for universal recovery (in which each node recovers all k packets). However, these conditions are numerous and cumbersome. The present paper gives a series of relatively simple conditions for universal recovery that apply when the number of packets is large and the distribution of packets among the nodes is well behaved. Our first result, which applies to any fixed network topology, uses only simple cuts in the network to characterize a set of transmission strategies such that for any ∈ > 0 these strategies require at most k∈ transmissions above the minimum required for universal recovery. For certain topologies including nonsingular d-regular d-connected networks, we explicitly construct transmission strategies that achieve universal recovery while using at most N transmissions above the minimum even when the total number of required transmissions is very large. These explicit constructions essentially resolve the problem completely for many canonical networks (e.g. cliques, rings, grids on tori, etc.).