Erasure Channel Research Articles

Optimal Memory Order of Memory-Based LT Encoders for Finite Block-Length Codes Over Binary Erasure Channels

Memory-based LT encoders (MBLTEs) have been shown to have better performance than the regular LT encoder in terms of bit error rate (BER) and decoding convergence speed. In this paper, we explore the entire family of MBLTEs for finite block-length codes over the binary erasure channel (BEC). We propose an algorithm to extend the first and second order MBLTE approach to an arbitrary i-th order MBLTE. We analyze the performance of such encoders mathematically by characterizing the expected accumulated number of recovered variable nodes at each decoding round. We define the threshold of the memory-based encoding method (MBEM) and show that the performance of MBLTEs increases as the memory order increases up to the point where this threshold is achieved. Beyond this point, we show that the performance of MBLTEs saturates if the channel erasure probability is zero and degrades otherwise. We formulate an optimization problem to solve for the optimal memory order based on whether or not the MBEM threshold is achieved. We present an extensive set of numerical results. These show agreement between our analysis and computer simulations. They also show that our optimization problem is efficient in determining the optimal memory order of MBLTEs in terms of decoding convergence speed, BER/frame-error-rate, and error floor.

Displacement Convexity in Spatially Coupled Scalar Recursions

We introduce a technique for the analysis of general spatially coupled systems that are governed by scalar recursions. Such systems can be expressed in variational form in terms of a potential function. We show, under mild conditions, that the potential function is displacement convex and that the minimizers are given by the fixed points (FPs) of the recursions. Furthermore, we give the conditions on the system such that the minimizing FP is unique up to translation along the spatial direction. The condition matches with that of Kudekar et al. [20] for the existence of spatial FPs. Displacement convexity applies to a wide range of spatially coupled recursions appearing in coding theory, compressive sensing, random constraint satisfaction problems, as well as statistical-mechanics models. We illustrate it with applications to low-density parity-check (LDPC) and generalized LDPC codes used for the transmission on the binary erasure channel or general binary memoryless symmetric channels within the Gaussian reciprocal channel approximation as well as compressive sensing.

OFDM Signal Recovery in Deep Faded Erasure Channel

In high-speed digital wireless communication applications, the intersymbol interference channel may have spectral nulls or erasures, which may degrade the orthogonal frequency division multiplexing (OFDM) bit error rate (BER) performance. In this paper, we prove that the BER performance of the OFDM system is considerably refined if the erased symbols in erasure channel (or received symbols on the deep-faded subcarriers in Rayleigh fading channel) are estimated based on the rest symbols in each OFDM signal. Also, an oversampled OFDM scheme which is a particular form of the pre-coded OFDM schemes is considered to guarantee high enough sampling rate. Furthermore, a new iterative receiver is proposed for perfect symbol recovery. A comprehensive Monte Carlo simulation study is performed to validate the theoretical results and affirm the superior performance of the oversampled OFDM scheme with the iterative receiver, compared with the typical detector for the OFDM scheme.

Performance Analysis and Design of LT Codes Under the Gaussian Elimination Algorithm in Wireless Sensor Networks

With the growing demand, wireless sensor networks (WSNs) play an increasingly important role in many fields. The focus has been on deploying Luby transform (LT) codes in WSNs because of their inherent advantages in erasure channels. Since most data transmitted by WSNs are small, the Gaussian elimination (GE) decoding algorithm is highly attractive. In this paper, we investigate the performance analysis and code design methods for LT codes under the GE algorithm. First, the analytical expressions of the data erasure probability are detailed for two simple kinds of LT codes. Then, a characteristic indicator of the data erasure probability is introduced to evaluate the performance of general LT codes. Finally, we propose novel design methods by optimizing degree distributions from three goals based on the characteristic indicator. The numerical results validate the proposed performance analysis and code design methods in comparison with the conventional method.

Analysis of Coding Strategies Within File Delivery Protocol Framework for HbbTV Based Push-VoD Services Over DVB Networks

Hybrid broadcast broadband TV is a technique providing Push-VoD services over an interactive hybrid TV. These services are broadcast using a file delivery protocol (FDP), which includes different coding strategies to ensure reliable delivery. This protocol is characterized by three levels of data representation giving rise to segmentation of packet losses, which may result in poor recovery capabilities. This paper provides a first thorough investigation of the coding FDP framework for reliable delivery of Push-VoD service over DVB networks. We propose Markov modeling for characterizing inter-layer loss propagation within FDP on a wide variety of burst erasure channels. Based on this analytical analysis and a simulation study, we determine the possible recovering areas and the accurate loss measurements within FDP. The latter is then used to effectively investigate and configure the different coding strategies provided within FDP. In addition, we present a suitable recovering strategy for FDP, which guarantees transmission robustness against the broadcast network impairments.

Protograph LDPC Codes With Block Thresholds: Extension to Degree-One and Generalized Nodes

Protograph low-density-parity-check (LDPC) codes are considered to design near-capacity low-rate codes over the binary erasure channel and the binary additive white Gaussian noise channel. For protographs with degree-one variable nodes and doubly-generalized LDPC (DGLDPC) codes, conditions are derived to ensure the equality of bit-error threshold and block-error threshold. Using this condition, low-rate codes with block-error threshold close to capacity are designed and shown to have better error rate performance than other existing codes.

Spatially Coupled Low-Density Parity-Check Codes on Two-Dimensional Array Erasure Channel

Spatially Coupled Low-Density Parity-Check Codes on Two-Dimensional Array Erasure Channel

On Privacy Amplification, Lossy Compression, and Their Duality to Channel Coding

We examine the task of privacy amplification from information-theoretic and coding-theoretic points of view. In the former, we give a one-shot characterization of the optimal rate of privacy amplification against classical adversaries in terms of the optimal type-II error in asymmetric hypothesis testing. This formulation can be easily computed to give finite-blocklength bounds and turns out to be equivalent to smooth min-entropy bounds by Renner and Wolf [Asiacrypt 2005] and Watanabe and Hayashi [ISIT 2013], as well as a bound in terms of the $E_\gamma$ divergence by Yang, Schaefer, and Poor [arXiv:1706.03866 [cs.IT]]. In the latter, we show that protocols for privacy amplification based on linear codes can be easily repurposed for channel simulation. Combined with known relations between channel simulation and lossy source coding, this implies that privacy amplification can be understood as a basic primitive for both channel simulation and lossy compression. Applied to symmetric channels or lossy compression settings, our construction leads to proto- cols of optimal rate in the asymptotic i.i.d. limit. Finally, appealing to the notion of channel duality recently detailed by us in [IEEE Trans. Info. Theory 64, 577 (2018)], we show that linear error-correcting codes for symmetric channels with quantum output can be transformed into linear lossy source coding schemes for classical variables arising from the dual channel. This explains a "curious duality" in these problems for the (self-dual) erasure channel observed by Martinian and Yedidia [Allerton 2003; arXiv:cs/0408008] and partly anticipates recent results on optimal lossy compression by polar and low-density generator matrix codes.

Secrecy Capacity of a Class of Erasure Wiretap Channels in WBAN.

In wireless body area networks (WBANs), the secrecy of personal health information is vulnerable to attacks due to the openness of wireless communication. In this paper, we study the security problem of WBANs, where there exists an attacker or eavesdropper who is able to observe data from part of sensors. The legitimate communication within the WBAN is modeled as a discrete memoryless channel (DMC) by establishing the secrecy capacity of a class of finite state Markov erasure wiretap channels. Meanwhile, the tapping of the eavesdropper is modeled as a finite-state Markov erasure channel (FSMEC). A pair of encoder and decoder are devised to make the eavesdropper have no knowledge of the source message, and enable the receiver to recover the source message with a small decoding error. It is proved that the secrecy capacity can be achieved by migrating the coding scheme for wiretap channel II with the noisy main channel. This method provides a new idea solving the secure problem of the internet of things (IoT).

Low-Latency Multichannel ALOHA With Fast Retrial for Machine-Type Communications

In this paper, for low-latency transmissions in machine-type communications, we study multichannel ALOHA with fast retrial that can shorten the access delay by immediate retransmissions of collided packets. To analyze the access delay of multichannel ALOHA with fast retrial, we employ an erasure channel model, which is equivalent to the widely used collision model, and study the effective capacity and the quality of service (QoS) exponent. Through the asymptotic analysis, we show that the QoS exponent grows logarithmically with the system dimension (i.e., the number of subchannels), which implies that multichannel ALOHA with fast retrial can effectively shorten the average access delay by increasing the system dimension. However, the growth rate of the effective capacity is shown to be slower than linear when the system dimension increases, which becomes the cost of improved access delay. In particular, the effective capacity is proportional to ${(N/\ln N)}$ , where ${N}$ is the number of subchannels. From the QoS exponent, the tail probability of queue is obtained and compared with simulation results, which confirms that the theoretical results from the derived QoS agree with the simulation results for large systems.

Overhead minimization of online codes

We introduce a novel method and two algorithms for designing efficient degree distributions with minimum overhead for online codes. The first algorithm is the general form for solving the problem of overhead minimization and the second one is an adaptive method based on the first method. In these two approaches, the codes are designed based on a discretized non-linear optimization problem. One direct result of the presented algorithms is decreasing the number of samples for solving the non-linear programming problem of the overhead minimization problem, which is not easy to solve without discretization. Also, we find a simple criteria in order to choose the critical sample points and show that the convergence rate improves. By considering these algorithms, one can construct an online code with minimum overhead for any given erasure channel parameter. Furthermore, the complexity of our algorithms has a linear relation with the number of samples because linear programming model is used. Simulation results are presented that show the efficiency of the proposed algorithms.

The Velocity of the Propagating Wave for Spatially Coupled Systems With Applications to LDPC Codes

We consider the dynamics of message passing for spatially coupled codes and, in particular, the set of density evolution equations that tracks the profile of decoding errors along the spatial direction of coupling. It is known that, for suitable boundary conditions and after a transient phase, the error profile exhibits a “solitonic behavior.” Namely, a uniquely shaped wavelike solution develops, which propagates with a constant velocity. Under this assumption, we derive an analytical formula for the velocity in the framework of a continuum limit of the spatially coupled system. The general formalism is developed for spatially coupled low-density parity-check codes on general binary memoryless symmetric channels, which form the main systems of interest in this paper. We apply the formula for special channels and illustrate that it matches the direct numerical evaluation of the velocity for a wide range of noise values. A possible application of the velocity formula to the evaluation of finite size scaling law parameters is also discussed. We conduct a similar analysis for general scalar systems and illustrate the findings with applications to compressive sensing and generalized low-density parity-check codes on the binary erasure or binary symmetric channels.

Calderbank-Shor-Steane holographic quantum error-correcting codes

We expand the class of holographic quantum error correcting codes by developing the notion of block perfect tensors, a wider class that includes previously defined perfect tensors. The relaxation of this constraint opens up a range of other holographic codes. We demonstrate this by introducing the self-dual CSS heptagon holographic code, based on the 7-qubit Steane code. Finally we show promising thresholds for the erasure channel by applying a straightforward, optimal erasure decoder to the heptagon code and benchmark it against existing holographic codes.

Robust quaternary fountain codes in AWGN interference

Luby transform (LT) codes have received less attention in larger non-binary fields. Robust soliton distribution (RSD) was designed to provide universal optimality over the binary erasure channel in terms of minimising the overhead. In this study, the authors introduce a fundamental degree distribution (FDD), which when used with non-binary LT codes, outperforms their binary counterparts using the RSD in terms of error rate and realised code rate. For instance, over an additive white Gaussian noise (AWGN) channel at of 12 dB and a rateless criteria (zero error probability), a non-systematic 4-ary LT code using the FDD can achieve a code rate of 0.75, compared to its binary counterpart which only achieves a code rate of 0.7.

Dephrasure Channel and Superadditivity of Coherent Information.

The quantum capacity of a quantum channel captures its capability for noiseless quantum communication. It lies at the heart of quantum information theory. Unfortunately, our poor understanding of nonadditivity of coherent information makes it hard to understand the quantum capacity of all but very special channels. In this Letter, we consider the dephrasure channel, which is the concatenation of a dephasing channel and an erasure channel. This very simple channel displays remarkably rich and exotic properties: we find nonadditivity of coherent information at the two-letter level, a substantial gap between the threshold for zero quantum capacity and zero single-letter coherent information, a big gap between single-letter coherent and private information, and positive quantum capacity for all complementary channels. Its clean form simplifies the evaluation of coherent information substantially and, as such, we hope that the dephrasure channel will provide a much-needed laboratory for the testing of new ideas about nonadditivity.

Improved online fountain codes

Online fountain codes have been proven to require lower overhead and fewer feedbacks than growth codes for successful decoding. In an attempt to improve the intermediate symbol recovery rate, the authors propose sending a number of degree-1 input symbols prior to the build-up phase based on a simple application of probability theory. In addition, during the completion phase, received encoded symbols with three neighbouring white (un-decoded) symbols are retained for decoding and updating the decoding graph later. The performance and characteristics of the proposed improved online fountain codes are compared to those of the original online fountain codes over an erasure channel. Simulation results reveal that the improved online fountain codes outperform the original fountain codes in terms of intermediate symbol recovery rate, average encoded symbols required to be generated by the sender, average feedback transmissions, and encoding/decoding efficiency.

Interleaver Design for Short Concatenated Codes

The choice of the interleaver may significantly affect the performance of short codes when they are used in serial concatenation. By focusing on the minimum distance of the overall concatenated code, we propose an algorithmic method for the design of good interleavers. As a valuable example of application, we consider the case of polar codes concatenated with cyclic redundancy check codes. For these codes, the method we propose is compared with classical approaches based on random searches to assess its advantages, which are also confirmed through examples of practical coded transmissions over the binary erasure channel.

Decoder Partitioning: Towards Practical List Decoding of Polar Codes

Polar codes represent one of the major recent breakthroughs in coding theory and, because of their attractive features, they have been selected for the incoming 5G standard. As such, a lot of attention has been devoted to the development of decoding algorithms with good error performance and efficient hardware implementation. One of the leading candidates in this regard is represented by successive-cancelation list (SCL) decoding. However, its hardware implementation requires a large amount of memory. Recently, a partitioned SCL (PSCL) decoder has been proposed to significantly reduce the memory consumption. In this paper, we consider the paradigm of PSCL decoding from a practical standpoint, and we provide several improvements. First, by changing the target signal-to-noise ratio and consequently modifying the construction of the code, we are able to improve the performance at no additional computational, latency, or memory cost. Second, we bridge the performance gap between SCL and PSCL decoding by introducing a generalized PSCL decoder and a layered PSCL decoder. In this way, we obtain almost the same performance of the SCL decoder with a significantly lower memory requirement, as testified by hardware implementation results. Third, we present an optimal scheme to allocate cyclic redundancy checks. Finally, we provide a lower bound on the list size that guarantees optimal maximum a posteriori performance for the binary erasure channel.

Estimation of Nonlinear Dynamic Systems Over Communication Channels

Remote observation of the state trajectory of nonlinear dynamic systems over limited capacity communication channels is studied. It is shown that two extreme cases are possible: Either the system is fully observable or the error in estimation blows up. The key observation is that such behavior is determined by the relationship between the Shannon capacity and the Lyapunov exponents; the well-known characterizing parameters of a communication channel on one side, and a dynamic system from the other side. In particular, it is proved that for nonlinear systems with initial state $x_0$ , the minimum capacity of an additive white Gaussian noise channel required for full observation of the system in the mean square sense is $ \sum _i\kappa _i(x_0)\Delta _i (x_0)$ , where $\Delta _i (x_0)$ s and $\kappa _i(x_0)$ s denote distinct Lyapunov exponents and their multiplicity numbers, respectively. Conversely, if the capacity is less than $E[\sum _i \kappa _i(x_0)\Delta _i (x_0)]$ , then observation is impossible. In order to show the universality of the result, we obtain the same observability conditions for the digital noiseless channel and the packet erasure channel in sure and almost sure senses, respectively.

Redundancy allocation in finite-length nested codes for nonvolatile memories

In this paper, we investigate the optimum way to allocate redundancy of finite-length nested codes for modern nonvolatile memories suffering from both permanent defects and transient errors (erasures or random errors). A nested coding approach such as partitioned codes can handle both permanent defects and transient errors by using two parts of redundancy: 1) redundancy to deal with permanent defects and 2) redundancy for transient errors. We consider two different channel models of the binary defect and erasure channel (BDEC) and the binary defect and symmetric channel (BDSC). The transient errors of the BDEC are erasures and the BDSC's transient errors are modeled by the binary symmetric channel, respectively. Asymptotically, the probability of recovery failure can converge to zero if the capacity region conditions of nested codes are satisfied. However, the probability of recovery failure of finite-length nested codes can be significantly variable for different redundancy allocations even though they all satisfy the capacity region conditions. Hence, we formulate the redundancy allocation problem of finite-length nested codes to minimize the recovery failure probability. We derive the upper bounds on the probability of recovery failure and use them to estimate the optimal redundancy allocation. Numerical results show that our estimated redundancy allocation matches well the optimal redundancy allocation.