Erasure Channel Research Articles

Stopping Sets of Algebraic Geometry Codes over Hyperelliptic Curves of Genus Two

Stopping sets are useful for analyzing the performance of a linear code under an iterative decoding algorithm over an erasure channel. In this paper, we consider stopping sets of one-point algebraic geometry codes defined by a hyperelliptic curve of genus g=2 defined by the plane model y2=f(x), where the degree of f(x) was 5. We completely classify the stopping sets of the one-point algebraic geometric codes C=CΩ(D,mP∞) defined by a hyperelliptic curve of genus 2 with m≤4. For m=3, we proved in detail that all sets S⊆{1,2,⋯,n} of a size greater than 3 are stopping sets and we give an example of sets of size 2,3 that are not.

Decoding error probability of random parity-check matrix ensemble over the erasure channel

AbstractIn this paper we carry out an in-depth study on the average decoding error probability of the random parity-check matrix ensemble over the erasure channel under three decoding principles, namely unambiguous decoding, maximum likelihood decoding and list decoding. We obtain explicit formulas for the average decoding error probabilities of the random parity-check matrix ensemble under these three decoding principles and compute the error exponents. Moreover, for unambiguous decoding, we compute the variance of the decoding error probability of the random parity-check matrix ensemble and the error exponent of the variance, which implies a strong concentration result, that is, roughly speaking, the ratio of the decoding error probability of a random linear code in the ensemble and the average decoding error probability of the ensemble converges to 1 with high probability when the code length goes to infinity.

The n-shot classical capacity of the quantum erasure channel

Abstract We compute the n-shot classical capacity of the quantum erasure channel, providing upper bounds and almost-matching lower bounds for it, the latter achievable via large-minimum-distance classical linear codes for any n. The protocols are in full product form, i.e. no entanglement is needed either at the encoder or decoder to attain the capacity, and they explicitly adapt to the transition between different error regimes as the erasure probability increases. Finally, we show that our upper and lower bounds on the capacity are tighter than those obtainable from the general theory of finite-size capacity via generalized divergences.

Classical product code constructions for quantum Calderbank-Shor-Steane codes

Several notions of code products are known in quantum error correction, such as hypergraph products, homological products, lifted products, balanced products, to name a few. In this paper we introduce a new product code construction which is a natural generalization of classical product codes to quantum codes: starting from a set of component Calderbank-Shor-Steane (CSS) codes, a larger CSS code is obtained where both X parity checks and Z parity checks are associated to classical product codes. We deduce several properties of product CSS codes from the properties of the component codes, including bounds to the code distance, and show that built-in redundancies in the parity checks result in so-called meta-checks which can be exploited to correct syndrome read-out errors. We then specialize to the case of single-parity-check (SPC) product codes which in the classical domain are a common choice for constructing product codes. Logical error rate simulations of a SPC 3-fold product CSS code having parameters [[512,174,8]] are shown under both a maximum likelihood decoder for the erasure channel and belief propagation decoding for depolarizing noise. We compare the results with other codes of comparable length and dimension, including a code from the family of asymptotically good Tanner codes. We observe that our reference product CSS code outperforms all the other examined codes.

Decoding of MDP Convolutional Codes over the Erasure Channel under Linear Systems Point of View

This paper attempts to highlight the decoding capabilities of MDP convolutional codes over the erasure channel by defining them as discrete linear dynamical systems, with which the controllability property and the observability characteristics of linear system theory can be applied, in particular those of output observability, easily described using matrix language. Those are viewed against the decoding capabilities of MDS block codes over the same channel. Not only is the time complexity better but the decoding capabilities are also increased with this approach because convolutional codes are more flexible in handling variable-length data streams than block codes, where they are fixed-length and less adaptable to varying data lengths without padding or other adjustments.

RL-ANC: Reinforcement Learning-Based Adaptive Network Coding in the Ocean Mobile Internet of Things

As the demand for sensing and monitoring the marine environment increases, the Ocean Mobile Internet of Things (OM-IoT) has gradually attracted the interest of researchers. However, the unreliability of communication links represents a significant challenge to data transmission in the OM-IoT, given the complex and dynamic nature of the marine environment, the mobility of nodes, and other factors. Consequently, it is necessary to enhance the reliability of underwater data transmission. To address this issue, this paper proposes a reinforcement learning-based adaptive network coding (RL-ANC) approach. Firstly, the channel conditions are estimated based on the reception acknowledgment, and a feedback-independent decoding state estimation method is proposed. Secondly, the sliding coding window is dynamically adjusted based on the estimates of the channel erasure probability and decoding probability, and the sliding rule is adaptively determined using a reinforcement learning algorithm and an enhanced greedy strategy. Subsequently, an adaptive optimization method for coding coefficients based on reinforcement learning is proposed to enhance the reliability of the underwater data transmission and underwater network coding while reducing the redundancy in the coding. Finally, the sampling period and time slot table are updated using the enhanced simulated annealing algorithm to optimize the accuracy and timeliness of the channel estimation. Simulation experiments demonstrate that the proposed method effectively enhances the data transmission reliability in unreliable communication links, improves the performance of underwater network coding in terms of the packet delivery rate, retransmission, and redundancy transmission ratios, and accelerates the convergence speed of the decoding probability.

Optimal Construction for Decoding 2D Convolutional Codes over an Erasure Channel

In general, the problem of building optimal convolutional codes under a certain criteria is hard, especially when size field restrictions are applied. In this paper, we confront the challenge of constructing an optimal 2D convolutional code when communicating over an erasure channel. We propose a general construction method for these codes. Specifically, we provide an optimal construction where the decoding method presented in the bibliography is considered.

Reed–Muller Codes on BMS Channels Achieve Vanishing Bit-Error Probability for all Rates Below Capacity

This paper considers the performance of Reed-Muller (RM) codes transmitted over binary memoryless symmetric (BMS) channels under bitwise maximum-a-posteriori (bit-MAP) decoding. Its main result is that, for a fixed BMS channel, the family of binary RM codes can achieve a vanishing bit-error probability at rates approaching the channel capacity. This partially resolves a long-standing open problem that connects information theory and error-correcting codes. In contrast with the earlier result for the binary erasure channel, the new proof does not rely on hypercontractivity. Instead, it combines a nesting property of RM codes with new information inequalities relating the generalized extrinsic information transfer function and the extrinsic minimum mean-squared error.

Age of Information Joint Optimization for an Energy Harvesting Network With Erasure Channel

Age of Information Joint Optimization for an Energy Harvesting Network With Erasure Channel

Stabilization and Optimal Control for Markov Jump Linear System With Two Controllers Over Multiple Erasure Channels

Stabilization and Optimal Control for Markov Jump Linear System With Two Controllers Over Multiple Erasure Channels

Movable Antenna-Aided Broadcast Packet Erasure Channels: Capacity with Dynamic Position Plan

Movable Antenna-Aided Broadcast Packet Erasure Channels: Capacity with Dynamic Position Plan

Coded Slotted ALOHA Scheme With Multi-Packet Reception Under Erasure Channels

Coded Slotted ALOHA Scheme With Multi-Packet Reception Under Erasure Channels

Event-Triggered Control with Intermittent Communications over Erasure Channels for Leader–Follower Problems with the Combined-Slip Effect

In this article, we investigate the vehicle path-following problem for a vehicle-to-vehicle (V2V)–enabled leader–follower scenario and propose an integrated control policy for the following vehicle to accurately follow the leader’s path. We propose a control strategy for the follower vehicle to maintain a velocity-dependent distance relative to the leader vehicle while stabilizing its longitudinal and lateral dynamics considering the combined-slip effect and tire force saturation. In light of reducing wireless communication errors and efficient usage of battery power and resources, we propose an intermittent V2V communication in which transmissions are scheduled based on an event-triggered law. An event is triggered and a transmission is scheduled in subsequent sample time if some of the well-defined path-following error functions (relative distance error and lateral error) exceed given tolerance bounds. Considering that the V2V communication channel might be erroneous or a transmission fails due to, e.g., vehicles’ distance or low battery power, we consider data loss in the V2V channel. Our proposed control law consists of two components: a receding horizon feedback controller with state constraints based on a safe operation envelop and a feedforward controller that generates complementary control inputs when the leader’s states are successfully communicated to the follower. To mitigate the effects of data loss on the follower’s path-following performance, we design a remote estimator for the follower to predict the leader’s state using its on-board sensor equipment when an event is triggered but the corresponding state information is not received by the follower due to a packet loss. Incorporating this estimator allows the follower to apply cautionary control inputs knowing that the path-following error had exceeded a tolerance bound. We show that while the feedback controller stabilizes the follower’s dynamics, the feedforward component improves the safety margins and reduces the path-following errors even in the presence of data loss. High-fidelity simulations are performed using CarSim to validate the effectiveness of our proposed control architecture specifically in harsh maneuvers and high-slip scenarios on various road surface conditions.

Quantum Full-Duplex Communication

Integrating the full-duplex capability with quantum communication potentially equips emerging wireless networks with a quantum layer of security for the stringent communication efficiency and security requirements. This paper proposes two new full-duplex quantum communication protocols to exchange classical or quantum information between two remote parties simultaneously without transferring a physical particle over the quantum channel. The first protocol, called <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">quantum duplex coding</i> , enables the exchange of a classical bit using a preshared maximally entangled pair of qubits by means of counterfactual disentanglement. The second protocol, called <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">quantum telexchanging</i> , enables the exchange of an arbitrary unknown qubit without using preshared entanglement by means of counterfactual entanglement and disentanglement. We demonstrate that quantum duplex coding and quantum telexchanging can be achieved by exploiting counterfactual electron-photon interaction gates. It is shown that these tasks can be viewed as full-duplex transmission of bits and qubits via binary erasure channels and quantum erasure channels, respectively.

Singleton Bounds for Entanglement-Assisted Classical and Quantum Error Correcting Codes

We show that entirely quantum Shannon theoretic methods, based on von Neumann entropies and their properties, can be used to derive Singleton bounds on the performance of entanglement-assisted hybrid classical-quantum (EACQ) error correcting codes. Concretely, we show that the triple-rate region of qubits, cbits and ebits of possible EACQ codes over arbitrary alphabet sizes is contained in the quantum Shannon theoretic rate region of an associated memoryless erasure channel, which turns out to be a polytope. We show that a large part of this region is attainable by certain EACQ codes, whenever the local alphabet size (i.e. Hilbert space dimension) is large enough, in keeping with known facts about classical and quantum maximum distance separable (MDS) codes: in particular, all of its extreme points and all but one of its extremal lines. The attainability of the remaining one extremal line segment is left as an open question.

Improving the Thresholds of Generalized LDPC Codes With Convolutional Code Constraints

CC-GLPDC codes are a class of generalized low-density parity-check (GLDPC) codes where the constraint nodes (CNs) represent convolutional codes. This allows for efficient decoding in the trellis with the forward-backward algorithm, and the strength of the component codes easily can be controlled by the encoder memory without changing the graph structure. In this letter, we extend the class of CC-GLDPC codes by introducing different types of irregularity at the CNs and investigating their effect on the BP and MAP decoding thresholds for the binary erasure channel (BEC). For the considered class of codes, an exhaustive grid search is performed to find the BP-optimized and MAP-optimized ensembles and compare their thresholds with the regular ensemble of the same design rate. The results show that irregularity can significantly improve the BP thresholds, whereas the thresholds of the MAP-optimized ensembles are only slightly different from the regular ensembles. Simulation results for the AWGN channel are presented as well and compared to the corresponding thresholds.

On the Required Radio Resources for Ultra-Reliable Communication in Highly Interfered Scenarios

Future wireless systems are expected to support mission-critical services demanding higher and higher reliability. In this letter, we dimension the radio resources needed to achieve a given failure probability target for ultra-reliable wireless systems in high interference conditions, assuming a protocol with frequency hopping combined with packet repetitions. We resort to packet erasure channel models and derive the minimum amount of resource units in the case of receiver with and without collision resolution capability, as well as the number of packet repetitions needed for achieving the failure probability target. Analytical results are numerically validated, and can be used as benchmark for realistic system simulations.

Generic Nonadditivity of Quantum Capacity in Simple Channels.

Determining capacities of quantum channels is a fundamental question in quantum information theory. Despite having rigorous coding theorems quantifying the flow of information across quantum channels, their capacities are poorly understood due to superadditivity effects. Studying these phenomena is important for deepening our understanding of quantum information, yet simple and clean examples of superadditive channels are scarce. Here we study a family of channels called platypus channels. Its simplest member, a qutrit channel, is shown to display superadditivity of coherent information when used jointly with a variety of qubit channels. Higher-dimensional family members display superadditivity of quantum capacity together with an erasure channel. Subject to the "spin-alignment conjecture" introduced in our companion paper [F. Leditzky, D. Leung, V. Siddhu, G. Smith, and J. A. Smolin, The platypus of the quantum channel zoo, IEEE Transactions on Information Theory (IEEE, 2023), 10.1109/TIT.2023.3245985], our results on superadditivity of quantum capacity extend to lower-dimensional channels as well as larger parameter ranges. In particular, superadditivity occurs between two weakly additive channels each with large capacity on their own, in stark contrast to previous results. Remarkably, a single, novel transmission strategy achieves superadditivity in all examples. Our results show that superadditivity is much more prevalent than previously thought. It can occur across a wide variety of channels, even when both participating channels have large quantum capacity.

A Novel LT Codec to Improve Error in the Binary Erasure Channel

A Novel LT Codec to Improve Error in the Binary Erasure Channel

Deep Reinforcement Learning for Latency-Sensitive Communication With Adaptive Redundant Retransmissions

This paper studies packet repetition strategies over erasure channels with memory and a long feedback delay. The problem is initially formulated as a communications problem where a source wishes to transmit one message packet to a destination while minimizing both the delay and the number of transmissions. At each time instant, the sender is provided a delayed acknowledgement feedback about past attempts, and must decide whether to attempt a new transmission or not. This problem is then re-formulated as an episodic reinforcement learning problem, where an agent attempts to learn the optimal transmission policy, provided delayed feedback about past transmission attempts. The agent is helped by a channel estimator, which attempts to capture the channel memory and use that to predict probabilities of erasures in a future window. This channel estimator is also data-driven and learns the channel model without any <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a priori</i> channel knowledge. The paper presents a lower bound on the achievable trade-off between delay and number of transmissions for any channel modeled as a Markov process. Experimental results show that the combination of the proposed channel estimator and the agent can noticeably outperform naive strategies for channels with memory, and achieves results close to the lower bound.