Erasure Channel Research Articles

Optimal Multiplexed Erasure Codes for Streaming Messages With Different Decoding Delays

This paper considers multiplexing two sequences of messages with two different decoding delays over a packet erasure channel. In each time slot, the source constructs a packet based on the current and previous messages and transmits the packet, which may be erased when the packet travels from the source to the destination. The destination must perfectly recover every source message in the first sequence subject to a decoding delay T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">v</sub> , and every source message in the second sequence subject to a shorter decoding delay T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u</sub> ≤ T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">v</sub> ,. We assume that the channel loss model introduces a burst erasure of a fixed length B on the discrete timeline. Under this channel loss assumption, the capacity region for the case where T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">v</sub> , ≤ T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u</sub> +B was previously solved. In this paper, we fully characterize the capacity region for the remaining case T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">v</sub> , ) T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u</sub> +B. The key step in the achievability proof is achieving the non-trivial corner point of the capacity region through using a multiplexed streaming code constructed by superimposing two single-stream codes. The main idea in the converse proof is obtaining a genie-aided bound when the channel is subject to a periodic erasure pattern where each period consists of a length-B burst erasure followed by a length-T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u</sub> noiseless duration.

On receiver diversity for grant-free based machine type communications

Grant-free access schemes are steadily gaining momentum for emerging machine type communications and internet of things applications. Relying on variations of the classical ALOHA protocol to allow a vast population of users share a common channel in an uncoordinated fashion, these solutions have recently been embedded in international standards as well as proprietary commercial products. In this context, the use of multiple low-complexity receivers that act as relays, collecting packets from users and forwarding them towards a central unit represents an interesting approach to increase the efficiency of the system, which finds natural application in both terrestrial and non-terrestrial settings. To gain insights on the potential of such a configuration, this paper investigates a two-tier slotted ALOHA system under an erasure channel model. Closed form expressions for both end-to-end throughput and packet loss rate are derived for an arbitrary number of receivers, and the role of correlation among wireless channels connecting users to relays is thoroughly discussed. Non-trivial tradeoffs and systems design hints are highlighted, pinpointing some fundamental benefits and limitations of receiver diversity for grant-free based machine type communications.

Bit-Error-Rate Analysis of Raptor Codes over Rician Fading Channels

The bit-error-rate expressions of nonsystematic Raptor (NR) codes and systematic Raptor (SR) codes over Rician fading channels are first derived using a Gaussian approximation (GA) approach. These BER expressions provide a significant reduction in computational complexity for analyzing system performance when compared with simulation and discretized density evolution (DDE). As shown by the asymptotic analysis, the NR codes originally designed for binary erasure channels still have good performance on Rician fading channels but SR codes do not. Therefore, the degree distributions of SR codes are specifically optimized on Rician channels which are superior to the existing SR codes and comparable to NR codes.

New coding scheme for the state estimation and reference tracking of nonlinear dynamic systems over the packet erasure channel (IoT): Applications in tele-operation of autonomous vehicles

This paper presents a new technique for the state estimation and reference tracking of nonlinear dynamic systems over the packet erasure channel, which is an abstract model for transmission via the Internet, WiFi wireless network and ZigBee modules. A new encoder and decoder for real time state estimation of nonlinear dynamic systems at the end of communication link when the measurements are sent through the limited capacity erasure channel, are presented. Then, using the available results from control theory, a controller for reference tracking and hence the stability of the system is also designed. That is, for nonlinear systems, almost sure asymptotic state estimation and reference tracking techniques including an encoder, decoder and a controller are presented. The satisfactory performances of the proposed state estimation and control techniques are illustrated via computer simulations by applying the proposed techniques on the unicycle model, which represents the dynamics of autonomous vehicles.

Optimal Source Codes for Timely Updates

A transmitter observing a sequence of independent and identically distributed random variables seeks to keep a receiver updated about its latest observations. The receiver need not be apprised about each symbol seen by the transmitter, but needs to output a symbol at each time instant t. If at time t the receiver outputs the symbol seen by the transmitter at time U (t) ≤ t, the age of information at the receiver at time t is t - U(t). We study the design of lossless source codes that enable transmission with minimum average age at the receiver. We show that the asymptotic minimum average age can be attained up to a constant gap by the Shannon codes for a tilted version of the original pmf generating the symbols, which can be computed easily by solving an optimization problem. Furthermore, we exhibit an example with alphabet X where age of a factor O(√(log |X|)) more than that achieved by our Shannon codes for the original pmf incur an asymptotic average codes. Underlying our prescription for optimal codes is a new variational formula for integer moments of random variables, which may be of independent interest. Also, we discuss possible extensions of our formulation to randomized schemes and to the erasure channel, and include a treatment of the related problem of source coding for minimum average queuing delay.

Energy-Aware Scheduling for Broadcast Erasure Channels With Two Energy Harvesting Receivers

A feedback-based coding algorithm is studied for broadcast packet erasure channels (BPECs) with energy harvesting receivers. A probabilistic algorithm is outlined, which is able to achieve the capacity region of two-user BPECs. The probabilistic algorithm is carefully studied from the optimization perspective. In this paper, we take into account the energy levels of receivers. Two new feedback schemes are proposed to reduce the energy consumption of receivers and we explain the advantage of each feedback scheme. We formulate the problem of maximizing the minimum weighted throughput of receivers for each feedback scheme. After several convexifications, we make the formulated optimizations tractable and propose an algorithm to solve the problems globally without compromising optimality. Once optimization problems are solved, we propose an energy-aware scheduling for packet broadcasting. Finally, a packet header design is proposed to enable utilization of the energy-aware scheduling. Simulation results verify that to achieve the optimal throughput, network-coding is necessary.

Overfitting effect of artificial neural network based nonlinear equalizer: from mathematical origin to transmission evolution

Overfitting effect of artificial neural network (ANN) based nonlinear equalizer (NLE) leads to a trap of bit error ratio (BER) overestimation in optical fiber communication system, especially when the performance is evaluated by the commonly-used pseudo-random binary sequence (PRBS). First, we mathematically investigate the PRBS generation and Gray code mapping rules, in comparison with the use of Mersenne Twister random sequence (MTRS). Under the condition of a symbol erasure channel, we identify that ANN can recognize both the PRBS generation and symbol mapping rules, by increasing the weights of NLE at specific positions, whereas the MTRS is currently safe owing to the limited input length of current ANN based NLE. Then, we design four channel models of fiber optical transmission to experimentally examine various impairments on the evolution of overfitting effect. When both the additive white Gaussian noise (AWGN) channel and the bandwidth limited channel are considered, the mitigation of overfitting becomes possible by the use of pruned PRBS (P-PRBS) training set with removing the generation and mapping rules determined input symbols. However, as for both the chromatic dispersion (CD) uncompensated channel and the CD managed channel, the overfitting effect becomes serious, because both CD and fiber nonlinearity induced inter-symbol interference (ISI) is beneficial for ANN to identify the PRBS symbol rules. Finally, possible solutions to mitigate the overfitting effect are summarized.

Towards systematic Luby transform codes: optimisation design over binary erasure channel

In this Letter, the authors introduce optimisation design technique of systematic Luby transform (SLT) codes over binary erasure channel (BEC). SLT codes are more efficient than Luby transform (LT) codes since the former ones can recover source symbols more quickly. There was some work about designing degree distribution in LT codes and it was shown that erasure probability barely impact performances of LT codes. Using degree distributions designed for LT codes directly in SLT codes does not provide satisfied performances. For the first time, they introduce erasure probability as a parameter to establish a linear programming model in SLT codes. By applying And–or Tree analysis, they propose the asymptotic analysis formulae of SLT codes over BEC. To achieve the goal of recovering source symbols as soon as possible, the authors' objective of optimisation is to minimise overhead. Simulations are offered to validate the performance of their optimisation result. Besides, their degree distribution outperforms other distributions with respect to bit error ratio.

A New Capacity-Approaching Scheme for General 1-to-K Broadcast Packet Erasure Channels With ACK/NACK

The capacity region of 1-to- K broadcast packet erasure channels with ACK/NACK is well known for some scenarios, e.g., $ {K}\leq 3$ , etc. However, existing achievability schemes either require knowing the target rate $\vec {{R}}$ in advance, and/or have a complicated description of the achievable rate region that is difficult to prove whether it matches the capacity or not. This work proposes a new network coding scheme with the following features: (i) its achievable rate region is identical to the capacity region for all the scenarios in which the capacity is known; (ii) its achievable rate region is much more tractable and has been used to derive new capacity rate vectors; (iii) it employs sequential encoding that naturally handles dynamic packet arrivals; (iv) it automatically adapts to unknown packet arrival rates $\vec {{R}}$ ; (v) it is based on $\mathsf {GF}({q})$ with ${q}\geq {K}$ . In addition to analytically characterizing the achievable rate region of the proposed scheme, numerical simulation has been used to verify its queue length and delay performance.

High‐parity vertical Reed‐Solomon codes for long GNSS high‐accuracy messages

This paper presents in detail an outer-coding proposal for long GNSS messages disseminated by multiple satellites. The proposal minimizes the message reception time, or Time To Retrieve Data (TTRD), over a page erasure channel in most types of environments and visibility conditions. A message M consisting of k pages is encoded into a set C consisting of n pages, such that when any subset C′ of C of k pages is correctly received, the message can be decoded. The scheme vertically encodes each page symbol in a separate high-parity Reed-Solomon code and allocates different parts of it to different satellites. The scheme has been designed for the Galileo High Accuracy Service (HAS), but it can be used for longer messages and other constellations. The performance results show TTRDs on the order of few seconds, at a very low decoding complexity for the receiver.

Recent Results on the Implementation of a Burst Error and Burst Erasure Channel Emulator Using an FPGA Architecture

The behaviour of a transmission channel may be simulated using the performance abilities of current generation multiprocessing hardware, namely, a multicore Central Processing Unit (CPU), a general purpose Graphics Processing Unit (GPU), or a Field Programmable Gate Array (FPGA). These were investigated by Cullinan et al. in a recent paper (published in 2012) where these three devices capabilities were compared to determine which device would be best suited towards which specific task. In particular, it was shown that, for the application which is objective of our work (i.e., for a transmission channel simulation), the FPGA is 26.67 times faster than the GPU and 10.76 times faster than the CPU. Motivated by these results, in this paper we propose and present a direct hardware emulation. In particular, a Cyclone II FPGA architecture is implemented to simulate a burst error channel behaviour, in which errors are clustered together, and a burst erasure channel behaviour, in which the erasures are clustered together. The results presented in the paper are valid for any FPGA architecture that may be considered for this scope.

Reconciliation for CV-QKD using globally-coupled LDPC codes**Project supported by the National Natural Science Foundation of China (Grant Nos. 61801522, 61972418, and 61872390), the Natural Science Foundation of Hunan Province, China (Grant Nos. 2019JJ40352 and 2017JJ3415), and the Special Foundation for Distinguished Young Scientists of Changsha City, China (Grant No. kq1905058).

Reconciliation is a necessary step in postprocessing of continuous-variable quantum key distribution (CV-QKD) system. We use globally coupled low-density parity-check (GC-LDPC) codes in reconciliation to extract a precise secret key from the raw keys over the authenticated classical public channel between two users. GC-LDPC codes have excellent performance over both the additive Gaussian white noise and binary-erasure channels. The reconciliation based on GC-LDPC codes can improve the reconciliation efficiency to 95.42% and reduce the frame error rate to 3.25 × 10–3. Using distillation, the decoding speed can achieve 23.8 Mbits/s and decrease the cost of memory. Given decoding speed and low memory usage, this makes the proposed reconciliation method viable approach for high-speed CV-QKD system.

A theoretical approach to a safety-based predictive adaptation of wireless communication channel parameters in harsh environments

This paper presents an approach to a real-time optimization of safety parameters in wireless communication systems. When considering the GEC–model (Generalized Erasure Channel) and the black channel design of a communication channel, then the PFH (Probability of Failure per Hour) value can be estimated using the parameters e – BER (bit-error-rate), φ – BLR (bit-loss-rate), v – number of safety related messages per second, n – message length and dmin – minimum distance of a linear code. The number of safety related messages per second v and the message length n, including the information block k and the checksum block r, can be varying between the permissible bounds. Accordingly, the variable parameters can be adjusted at run-time with additional assimilation of the used cyclic code. It allows the real-time prediction and optimization of the safety parameters. In this paper, the concept of the parameter estimation is discussed and based on it the optimization problem is defined.

Multi-Erasure Locally Recoverable Codes Over Small Fields: A Tensor Product Approach

Erasure codes play an important role in storage systems to prevent data loss. In this work, we study a class of erasure codes called Multi-Erasure Locally Recoverable Codes (ME-LRCs) for storage arrays. Compared to previous related works, we focus on the construction of ME-LRCs over small fields. Our main contribution is a general construction of ME-LRCs based on generalized tensor product codes, and an analysis of their erasure-correcting properties. A decoding algorithm tailored for erasure recovery is given, and correctable erasure patterns are identified. We then prove that our construction yields optimal ME-LRCs with a wide range of code parameters, and present some explicit ME-LRCs over small fields. Next, we show that generalized integrated interleaving (GII) codes can be treated as a subclass of generalized tensor product codes, thus defining the exact relation between these codes. Finally, ME-LRCs are investigated in a probabilistic setting. We prove that ME-LRCs based upon a generalized tensor product construction can achieve the capacity of a compound erasure channel consisting of a family of erasure product channels.

New Two-Stage Automorphism Group Decoders for Cyclic Codes

Recently, error correcting codes in the erasure channel have drawn great attention for various applications such as distributed storage systems and wireless sensor networks, but many of their decoding algorithms are not practical because they have higher decoding complexity and longer delay. Thus, the automorphism group decoder (AGD) for cyclic codes in the erasure channel was introduced, which has good erasure decoding performance with low decoding complexity. In this paper, we propose new two-stage AGDs (TS-AGDs) for cyclic codes in the erasure channel by modifying the parity-check matrix and introducing the preprocessing stage to the AGD scheme. The proposed TS-AGD is analyzed for binary extended Golay and BCH codes. Also, TS-AGD can be used in the error channel using ordered statistics. Through numerical analysis, it is shown that the proposed decoding algorithm has good erasure decoding performance with lower decoding complexity than the conventional AGD. For some cyclic codes, it is shown that the proposed TS-AGD achieves the performance nearly identical to the maximum likelihood (ML) decoder in the erasure channel and the ordered statistics decoder (OSD) in the error channel.

Centralized vs Decentralized Targeted Brute-Force Attacks: Guessing With Side-Information

According to recent empirical studies, a majority of users have the same, or very similar, passwords across multiple password-secured online services. This practice can have disastrous consequences, as one password being compromised puts all the other accounts at much higher risk. Generally, an adversary may use any side-information he/she possesses about the user, be it demographic information, password reuse on a previously compromised account, or any other relevant information to devise a better brute-force strategy (so called targeted attack). In this work, we consider a distributed brute-force attack scenario in which $m$ adversaries, each observing some side information, attempt breaching a password secured system. We compare two strategies: an uncoordinated attack in which the adversaries query the system based on their own side-information until they find the correct password, and a fully coordinated attack in which the adversaries pool their side-information and query the system together. For passwords X of length $n$ , generated independently and identically from a distribution $P_{X}$ , we establish an asymptotic closed-form expression for the uncoordinated and coordinated strategies when the side-information $\mathrm {Y}_{(m)}$ are generated independently from passing X through a memoryless channel $P_{Y|X}$ , as the length of the password $n$ goes to infinity. We illustrate our results for binary symmetric channels and binary erasure channels, two families of side-information channels which model password reuse. We demonstrate that two coordinated agents perform asymptotically better than any finite number of uncoordinated agents for these channels, meaning that sharing side-information is very valuable in distributed attacks.

Research of LT Code Based on Key Information Feedback in Deep Space Communication

CCSDS LDPC code cannot effectively solve the problems of communication interruptions and high communication error rates in deep space communication. An application scheme of digital fountain code for deep space communication is thoroughly studied in this paper. First, an improved joint coding and decoding algorithm for LT code is proposed by adjusting the probability of the source symbol selection to achieve a nonuniform selection coding algorithm and by introducing the idea of real-time decoding on the basis of combining the BP and GE decoding algorithms to disperse the decoding complexity. Second, to ensure that the LT code can be reliably transmitted under different channel conditions, a feedback information extraction method and a retransmission coding method are designed, and an LT code transmission scheme based on key information feedback is proposed. Finally, the concatenation of the LDPC code with the LT code based on key information feedback is proposed. The LDPC code processing can be equivalently run on deep-space channels as erasure channels so that the advantage of the LT code being rateless can still be exploited in deep-space communication. Through simulation experiments, the performance of the cascade codes is analyzed from the two aspects of decoding efficiency and decoding overhead, and the performance of the cascade codes and CCSDS LDPC codes are compared from the perspective of the bit error rate. Simulation results show that the decoding efficiency of the cascade code increases, the decoding overhead decreases, and the communication bit error rate is lower in deep space communication compared with the CCSDS LDPC code. Therefore, the cascade code can improve the reliability of deep space communication.

Guessing with a Bit of Help.

What is the value of just a few bits to a guesser? We study this problem in a setup where Alice wishes to guess an independent and identically distributed (i.i.d.) random vector and can procure a fixed number of k information bits from Bob, who has observed this vector through a memoryless channel. We are interested in the guessing ratio, which we define as the ratio of Alice’s guessing-moments with and without observing Bob’s bits. For the case of a uniform binary vector observed through a binary symmetric channel, we provide two upper bounds on the guessing ratio by analyzing the performance of the dictator (for general ) and majority functions (for ). We further provide a lower bound via maximum entropy (for general ) and a lower bound based on Fourier-analytic/hypercontractivity arguments (for ). We then extend our maximum entropy argument to give a lower bound on the guessing ratio for a general channel with a binary uniform input that is expressed using the strong data-processing inequality constant of the reverse channel. We compute this bound for the binary erasure channel and conjecture that greedy dictator functions achieve the optimal guessing ratio.

Exploring the Limitations of Quantum Networking through Butterfly‐Based Networks

AbstractThe classical and quantum networking regimes of the butterfly network and a group of larger networks constructed with butterfly network blocks are investigated. By considering simultaneous multicasts from a set of senders to a set of receivers, the corresponding rates for transmitting classical and quantum information through the networks are analyzed. More precisely, achievable rates (i.e., lower bounds) for classical communication are compared with upper bounds for quantum communication, quantifying the performance gap between the rates for networks connected by identity, depolarizing, and erasure channels. For each network considered, a range is observed over which the classical rate non‐trivially exceeds the quantum capacity. It is found that, by adding butterfly blocks in parallel, the difference between transmitted bits and qubits can be increased up to one extra bit per receiver in the case of perfect transmission (identity channels). The aim is to provide a quantitative analysis of those network configurations which are particularly disadvantageous for quantum networking, when compared to classical communication. By clarifying the performance of these “negative cases,” some guidance on how quantum networks should be built is also provided.

Tail-Biting Globally-Coupled LDPC Codes

This paper presents a new type of globally-coupled low-density parity-check (GC-LDPC) codes whose base matrix has a cyclic structure in the global part. Therefore, the resulting codes are referred to as tail-biting GC-LDPC (TB-GC-LDPC) codes. We propose two methods to construct TB GC quasi-cyclic LDPC (TB-GC-QC-LDPC) codes. For the first method, we extract a replicated version of a constructed base matrix and mask it with a designed masking matrix. Compared to the conventional construction methods, this method provides more flexibility in code length for TB-GC-QC-LDPC codes. The second method is based on designing the incidence matrix of a special type of packings. Examples show that the constructed TB-GC-QC-LDPC codes perform well over the additive white Gaussian noise channel (AWGNC) and the binary erasure channel (BEC). The asymptotic performance of TB-GC-LDPC ensembles over BECs are also analyzed by resorting to density evolution. Moreover, numerical results show that TB-GC-LDPC ensembles can achieve better flooding-schedule decoding (FSD) thresholds than the corresponding GC-LDPC ensembles with a similar structure. With sufficient decoding iterations in the global phase, the local/global two-phase iterative decoding (TPD) thresholds of the TB-GC-LDPC ensembles significantly outperform those of the corresponding GC-LDPC ensembles as well.