Erasure Probability Research Articles

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.

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.

Demonstrating a Long-Coherence Dual-Rail Erasure Qubit Using Tunable Transmons

Quantum error correction with erasure qubits promises significant advantages over standard error correction due to favorable thresholds for erasure errors. To realize this advantage in practice requires a qubit for which nearly all errors are such erasure errors, and the ability to check for erasure errors without dephasing the qubit. We demonstrate that a “dual-rail qubit” consisting of a pair of resonantly coupled transmons can form a highly coherent erasure qubit, where transmon T1 errors are converted into erasure errors and residual dephasing is strongly suppressed, leading to millisecond-scale coherence within the qubit subspace. We show that single-qubit gates are limited primarily by erasure errors, with erasure probability perasure=2.19(2)×10−3 per gate while the residual errors are ∼40 times lower. We further demonstrate midcircuit detection of erasure errors while introducing <0.1% dephasing error per check. Finally, we show that the suppression of transmon noise allows this dual-rail qubit to preserve high coherence over a broad tunable operating range, offering an improved capacity to avoid frequency collisions. This work establishes transmon-based dual-rail qubits as an attractive building block for hardware-efficient quantum error correction. Published by the American Physical Society 2024

Energy and MCS Optimization in HARQ Protocol for Ultrareliable Regime With Maximized Throughput

In this article, we propose a cross-layer approach to maximize the throughput efficiency of the hybrid automatic repeat request (HARQ) protocol while guaranteeing ultrareliable communication (URC). The study combines adaptive modulation and coding and power control, at the physical layer, with automatic repeat request (ARQ)/HARQ at the data-link layer. The proposed optimization approach, based on genetic algorithms, allows a judicious allocation of the available energy budget and a suitable modulation and coding scheme (MCS) choice while keeping the same energy budget. For a given signal-to-noise ratio (SNR), a vector of energy and an MCS are modeled as one individual. The individuals with the highest throughput efficiency are selected to be used in generating the next population through crossover of the survivals and mutation. The individual with the highest throughput efficiency in the last population presents the solution offering the optimized parameters used in the packet transmission. We compare our solution to an existing one that exploits Lagrange multipliers. The results show that the proposed genetic algorithm based solution ensures URC with packet erasure probability less than <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$10^{-5}$</tex-math></inline-formula> from SNR values of 7 and 13 dB in static and fading channels, respectively.

Erasure of strings and vortices

The interaction of defects can lead to a phenomenon of erasure. During this process, a lower-dimensional object gets absorbed and dissolved by a higher-dimensional one. The phenomenon is very general and has a wide range of implications, both cosmological and fundamental. In particular, all types of strings, such as cosmic strings, QCD flux tubes, or fundamental strings, get erased when encountering a defect, either solitonic or a D-brane that deconfines their fluxes. This leads to a novel mechanism of cosmic string breakup, accompanied by gravitational and electromagnetic radiations. The arguments based on loss of coherence and the entropy count suggest that the erasure probability is very close to one, and strings never make it through the deconfining layer. We confirm this by a numerical simulation of the system, which effectively captures the essence of the phenomenon: a $2+1$-dimensional problem of interaction between a Nielsen-Olesen vortex of a $U(1)$ Higgs model and a domain wall, inside which the $U(1)$ gauge group is un-Higgsed and the magnetic flux is deconfined. In accordance with the entropy argument, in our simulation, the vortex never makes it across the wall.

Performance Bounds of Coded Slotted ALOHA Over Erasure Channels

In this paper, we derive an upper bound on load-threshold of coded slotted ALOHA (CSA) for slot erasure channel (SEC) and that for packet erasure channel (PEC), respectively. In SEC, by exploiting the analogy between Gaussian elimination of solving linear equations and the iterative successive interference-cancellation (SIC) decoding procedure of CSA, we derive the upper performance bound for SEC. Subsequently, in PEC, utilizing density evolution to track the average probability of packet-decoded failure within the SIC decoding process, we obtain the upper bound for PEC. Moreover, we propose a numerical search algorithm of these bounds under a given calculating precision. The numerical results indicate that the asymptotic upper limit on load-threshold of CSA over SEC is always less than that over PEC with any given erasure probability and average transmission rate.

Weighted zigzag decodable fountain codes for unequal error protection

By combining bit-shift and exclusive-or operations, a weighted zigzag decodable fountain code is proposed to achieve an unequal error protection property. In the proposed scheme, the input symbols of different importance levels are first pre-coded into variable nodes using low-density parity-check codes. Then, bit-shift operations prior to exclusive-or are performed on the non-uniformly selected variable nodes to generate encoded symbols. An analysis of the erasure probabilities based on the and-or tree is further introduced. Simulation results show that by appropriately choosing the maximum bit-shift amount, the proposed weighted zigzag decodable fountain codes can successfully recover the more important input symbols prior to the less important ones.

Timely Status Updating Over Erasure Channels Using an Energy Harvesting Sensor: Single and Multiple Sources

A status updating system is considered in which data from multiple sources are sampled by an energy harvesting sensor and transmitted to a remote destination through an erasure channel. The goal is to deliver status updates of all sources in a timely manner, such that the cumulative long-term average <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">age-of-information</i> (AoI) is minimized. The AoI for each source is defined as the time elapsed since the generation time of the latest successful status update received at the destination from that source. Transmissions are subject to energy availability, which arrives in units according to a Poisson process, with each energy unit capable of carrying out one transmission from only one source. The sensor is equipped with a unit-sized battery to save the incoming energy. A scheduling policy is designed in order to determine which source is sampled using the available energy. The problem is studied in two main settings: 1) no erasure status feedback and 2) perfect instantaneous feedback. For the case of one source, it is shown that <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">renewal policies</i> are optimal, in which successful status update instances form a renewal process. In the setting without feedback, it is further shown that <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">threshold-based policies</i> are optimal, in which the source is sampled only if the time until a new energy unit arrives exceeds a certain threshold. In the setting with feedback, threshold-greedy policies are investigated, in which the source is sampled according to a threshold-based policy following successful transmissions, and instantaneously whenever energy is available following failed transmissions. The optimal thresholds are found in closed-form in terms of the erasure probability. Such threshold-based policies are then extended for the case of multiple sources, combined with <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">round robin</i> scheduling in the setting without feedback, in which sources are sampled in the same repeating order; and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">maximum-age-first</i> scheduling in the setting with feedback, in which sources with maximum AoI are given priority. In both settings, the achieved cumulative long-term average AoI is derived in closed-form in terms of the threshold, the erasure probability and the number of sources.

Coded Wireless Distributed Computing With Packet Losses and Retransmissions

In wireless distributed computing systems, mobile devices that are connected wirelessly to the Fog (e.g., small base stations) collaboratively solve a given computational task. Unfortunately, wireless distributed computing systems suffer from packet losses due to severe channel fading. Moreover, a wireless device can drop out of the system when leaving the coverage of a master node in the Fog layer. We model this unreliability between a device and a master node as a packet erasure channel. When a packet fails to be detected at the receiver, the corresponding packet is retransmitted, which would significantly increase the overall run-time to finish the task. We take a coding-theoretic approach to tackle this straggler-like problem in wireless distributed computing. We first investigate the expected latency using an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(n, k)$ </tex-math></inline-formula> maximum-distance separable (MDS) code. We obtain the lower and upper bounds on the latency in closed forms and provide guidelines to design MDS codes depending on the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">channel</i> condition characterized by packet erasure probability. Then, we introduce another important performance metric called <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">minimum latency</i> , and also provide guidelines on designing optimal codes. Based on optimal codes, we obtain the performance curves of achievable minimum latency and achievable workload as functions of packet erasure probability.

Direct-to-Satellite IoT Slotted Aloha Systems with Multiple Satellites and Unequal Erasure Probabilities.

Direct-to-satellite Internet of Things (IoT) solutions have attracted a lot of attention from industry and academia recently, as promising alternatives for large scale coverage of a massive number of IoT devices. In this work, we considered that a cluster of IoT devices was under the coverage of a constellation of low-Earth orbit (LEO) satellites, while slotted Aloha was used as a medium access control technique. Then, we analyzed the throughput and packet loss rate while considering potentially different erasure probabilities at each of the visible satellites within the constellation. We show that different combinations of erasure probabilities at the LEO satellites and the IoT traffic load can lead to considerable differences in the system’s performance. Next, we introduce an intelligent traffic load distribution (ITLD) strategy, which, by choosing between a non-uniform allocation and the uniform traffic load distribution, guarantees a high overall system throughput, by allocating more appropriate amounts of traffic load at different positions (i.e., different sets of erasure probabilities) of the LEO constellation with respect to the IoT cluster. Finally, the results show that ITLD, a mechanism with low implementation complexity, allows the system to be much more scalable, intelligently exploiting the potential of the different positions of the satellite constellation.

Performance Analysis of Feedback-Based Network-Coded Systems for Broadcast

In a broadcast transmission over an erasure network, feedback with network coding is both capacity-achieving and loss-immune. In this article, we carry out performance analyses of the feedback-based online network coding technique, drop when seen (DWS), and its recently proposed randomized variant (rDWS). An absorbing discrete time Markov chain (DTMC) model is constructed to find the probability of dropping a packet from the sender queue (even before decoding) in a generation-based DWS broadcast. The dropping probability is computed efficiently in homogeneous as well as in heterogeneous erasure scenario by a tensor-algebraic formulation on top of the DTMC model. Further, the notion of first time drop leads to obtain a dropping distribution. A similar inspection for rDWS is not straight-forward because of receivers’ Markov-state dependency. So, the analysis is carried out with respect to Independent Markov-state Model (IMM), and the model is shown to be accurate in most of the cases except the situations where erasure probabilities, number of receivers, field size, target time slot all are very low. Another characterization of the dropping phenomenon helps in establishing the fact, average dropping time strictly decreases with generation size for DWS , and we conjecture, the same holds for rDWS. Using the fundamental matrix concept, we calculate the mean decoding time of a generation for a receiver. Extending the idea of first time dropping to decoding, statistics of different decoding options are analyzed. Finally, some analytical and simulation plots yield, the light-weight rDWS exhibits a very close performance to DWS for a sufficiently large finite field.

Analytic Expressions of Decoding Thresholds for LDPC Codes Over BEC

Analytic expressions of decoding thresholds both for regular and irregular low-density parity-check (LDPC) codes over the binary erasure channel (BEC) are given, and low-complexity methods for determining the thresholds are proposed based on the analytic expressions. Specifically, firstly, a fixed-point equation, which represents a constraint relationship between the threshold and erasure probability of transmitted bits, is established. Secondly, by introducing an auxiliary equation, the threshold can be solved as an analytically function of solutions of a polynomial equation. Moreover, this method is extended to the Additive White Gaussian Noise (AWGN) channel. Numerical results show that, our proposed method is more accurate, and the complexity is much lower than that of conventional density evolution (DE). Moreover, our method does not require to compute the inverse of degree distribution function for irregular LDPC codes.

Control of Two Satellites Relative Motion over the Packet Erasure Communication Channel with Limited Transmission Rate Based on Adaptive Coder

The paper deals with the navigation data exchange between two satellites moving in a swarm. It is focused on the reduction of the inter-satellite demanded communication channel capacity taking into account the dynamics of the satellites relative motion and possible erasures in the channel navigation data. The feedback control law is designed ensuring the regulation of the relative satellites motion. The adaptive binary coding/decoding procedure for the satellites navigation data transmission over the limited capacity communication channel is proposed and studied for the cases of ideal and erasure channels. Results of the numerical study of the closed-loop system performance and accuracy of the data transmission algorithm on the communication channel bitrate and erasure probability are obtained by extensive simulations. It is shown that both data transmission error and regulation time depend approximately inversely proportionally on the communication rate. In addition the erasure of data in the channel with probability up to 0.3 does not influence the regulation time for sufficiently high data transmission rate.

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.

On the Probability of Erasure for MIMO-OFDM

Detecting the presence of a valid signal is an important task of a telecommunication receiver. When the receiver is unable to detect the presence ofa valid signal, due to noise and fading, it is referred to as an erasure. Thiswork deals with the probability of erasure computation for orthogonalfrequency division multiplexed (OFDM) signals used by multiple inputmultiple output (MIMO) systems. The theoretical results are validated bycomputer simulations. OFDM is widely used in present day wireless communication systems due to its ability to mitigate intersymbol interference(ISI) caused by frequency selective fading channels. MIMO systems offerthe advantage of spatial multiplexing, resulting in increased bit-rate, whichis the main requirement of the recent wireless standards like 5G and beyond.

Application of the information-percolation method to reconstruction problems on graphs

In this paper we propose a method of proving impossibility results based on applying strong data-processing inequalities to estimate mutual information between sets of variables forming certain Markov random fields. The end result is that mutual information between two “far away” (as measured by the graph distance) variables is bounded by the probability of the existence of an open path in a bond-percolation problem on the same graph. Furthermore, stronger bounds can be obtained by establishing mutual information comparison results with an erasure model on the same graph, with erasure probabilities given by the contraction coefficients. As applications, we show that our method gives sharp threshold for partially recovering a rank-one perturbation of a random Gaussian matrix (spiked Wigner model), yields the best known upper bound on the noise level for group synchronization (obtained concurrently by Abbe and Boix), and establishes new impossibility result for community detection on the stochastic block model with k communities.

Polylog-LDPC Capacity Achieving Codes for the Noisy Quantum Erasure Channel

We provide polylog sparse quantum codes for correcting the Erasure channel arbitrarily close to the capacity. Specifically, we provide the $[[n, k, d]]$ quantum stabilizer codes that correct for the erasure channel arbitrarily close to the capacity if the erasure probability is at least 0.33 and with a generating set $\langle S_{1}, S_{2}, \cdots {} S_{n-k} \rangle $ such that $|S_{i}|\leq {\mathrm{ log}} ^{2+\zeta }(n)$ for all $i$ and for any $\zeta > 0$ with high probability. In this paper, we show that the result of Delfosse et al. is tight: one can construct capacity approaching codes with weight almost $O(1)$ .

Generalized Partial Orders for Polar Code Bit-Channels

We study partial orders (POs) for the synthesized bit-channels of polar codes. First, we give an alternative proof of an existing PO for bit-channels with the same Hamming weight and use the underlying idea to extend the bit-channel ordering to some additional cases. In particular, the bit-channel ordering for a given code block length is used to generate additional bit-channel ordering relationships for larger block lengths, generalizing previously known POs. Next, we consider POs especially for the binary erasure channel (BEC). We identify a symmetry property of the Bhattacharyya parameters of complementary bit-channel pairs on the BEC and provide a condition for the alignment of polarized sets of bit-channels for the BEC and general binary-input memoryless symmetric (BMS) channels. Numerical examples and further properties about the POs for the bit-channels with different Hamming weights are provided to illustrate the new POs. The bit-channels with universal ordering positions, which are independent of the channel erasure probability, are verified for all of the code block lengths. Finally, we show the threshold behavior of the Bhattacharyya parameters of some bit-channels by approximating the threshold values. The corresponding value for a bit-channel can be used to determine whether it is good or bad when the underlying channel is known.

Studying the Binary Erasure Polarization Subchannels Using Network Reliability

In this article we analyze the reliability of the synthetic channels of a polar code designed for the binary erasure channel, and in particular the properties of their Bhattacharyya parameter. We apply techniques from network theory in order to establish fresh results. Firstly, we show how to efficiently approximate the Bhattacharyya parameter. Secondly, we identify asymptotically “good” channels for polar code. Thirdly, we determine a range of values for the erasure probability for which the selecting rule for the polar codes construction coincides with that of the Reed-Muller codes.

Spatially coupled LDPC-BCH codes in quantum secure direct communications

Quantum secure direct communication systems are limited by high erasure probabilities and low receiving probabilities due to the high light source losses in the optical elements and optical fibers. A channel mitigation method based on spatially coupled LDPC-BCH codes is presented here to ensure reliable information transformations in quantum secure direct communication systems to resolve the problems of low-rate and Shannon limit-approaching code designs. Spatially coupled LDPC-BCH codes are constructed by coupling multiple identical block LDPC-BCH codes with edge spreading. The extrinsic information transfer functions of the spatially coupled LDPC-BCH code are analyzed to derive the decoding thresholds for the code ensembles. Simulations show that the spatially coupled LDPC-BCH codes have decoding thresholds closer to the Shannon limit with lower bit error rates than block LDPC-BCH codes. The spatially coupled LDPC-BCH code channel mitigation scheme has a transmitting efficiency of 0.001 and a bit error rate less than 10 −6 when the channel receiving probability is set at 0.4%. 摘要 受光源、光器件和光纤损耗影响, 量子安全直接通信系统的信息传输面临删除概率极大、接收概率极低的难题。为保证量子安全直接通信系统信息传输的可靠性, 该文提出了基于空间耦合LDPC-BCH码的信道补偿方案, 解决了极低码率码字构造和逼近Shannon限实用码字设计2方面的问题。将多个相同的分组LDPC-BCH码通过边连接的方式耦合起来构造得到了空间耦合LDPC-BCH码。建立了空间耦合LDPC-BCH码集的外信息转移函数, 并基于外信息转移函数分析了码集的译码门限值。仿真结果表明：与分组LDPC-BCH码相比, 空间耦合LDPC-BCH码的译码门限更接近Shannon限, 误比特率更低。在信道接收概率为0.4%时, 采用传输效率为0.001的空间耦合LDPC-BCH码信道补偿方案, 系统误比特率低于10 -6 。