Random Erasures Research Articles

Multiplexed Streaming Codes for Messages With Different Decoding Delays in Channel with Burst and Random Erasures

Multiplexed Streaming Codes for Messages With Different Decoding Delays in Channel with Burst and Random Erasures

Distributed Chernoff Test: Optimal Decision Systems Over Networks

We study “active” decision making over sensor networks where the sensors' sequential probing actions are actively chosen by continuously learning from past observations. We consider two network settings: with and without central coordination. In the first case, the network nodes interact with each other through a central entity, which plays the role of a fusion center. In the second case, the network nodes interact in a fully distributed fashion. In both of these scenarios, we propose sequential and adaptive hypothesis tests extending the classic Chernoff test. We compare the performance of the proposed tests to the optimal sequential test. In the presence of a fusion center, our test achieves the same asymptotic optimality of the Chernoff test, minimizing the risk, expressed by the expected cost required to reach a decision plus the expected cost of making a wrong decision, when the observation cost per unit time tends to zero. The test is also asymptotically optimal in the higher moments of the time required to reach a decision. Additionally, the test is parsimonious in terms of communications, and the expected number of channel uses per network node tends to a small constant. In the distributed setup, our test achieves the same asymptotic optimality of Chernoff's test, up to a multiplicative constant in terms of both risk and the higher moments of the decision time. Additionally, the test is parsimonious in terms of communications in comparison to state-of-the-art schemes proposed in the literature. The analysis of these tests is also extended to account for message quantization and communication over channels with random erasures.

Rate-Optimal Streaming Codes for Channels With Burst and Random Erasures

In this paper, we design erasure-correcting codes for channels with burst and random erasures, when a strict decoding delay constraint is in place. We consider the sliding-window-based packet erasure model proposed by Badr et al., where any time-window of width $w$ contains either up to $a$ random erasures or an erasure burst of length at most $b$ . One needs to recover any erased packet with a strict decoding delay deadline of $\tau $ , where erasures are as per the channel model. Presently existing rate-optimal constructions in the literature require, in general, a field-size which grows exponential in $\tau $ , as long as $\frac {a}{\tau }$ remains a constant. In this work, we present a new rate-optimal code construction covering all channel and delay parameters, which requires an $O(\tau ^{2})$ field-size. As a special case, when $(b-a)=1$ , we have a field-size linear in $\tau $ . We also present two other constructions having linear field-size, under certain constraints on channel and decoding delay parameters. As a corollary, we obtain low field-size, rate-optimal convolutional codes for any given column distance and column span. Simulations indicate that the newly proposed streaming code constructions offer lower packet-loss probabilities compared to existing schemes, for selected instances of Gilbert-Elliott and Fritchman channels.

Efficiently Decoding Reed–Muller Codes From Random Errors

Reed–Muller (RM) codes encode an $m$ -variate polynomial of degree at most $r$ by evaluating it on all points in $\{0,1\}^{m}$ . We denote this code by $RM(r,m)$ . The minimum distance of $RM(r,m)$ is $2^{m-r}$ and so it cannot correct more than half that number of errors in the worst case. For random errors one may hope for a better result. In this paper we give an efficient algorithm (in the block length $n=2^{m}$ ) for decoding random errors in RM codes far beyond the minimum distance. Specifically, for low-rate codes (of degree $r=o(\sqrt {m})$ ), we can correct a random set of $(1/2-o(1))n$ errors with high probability. For high rate codes (of degree $m-r$ for $r=o(\sqrt {m/\log m})$ ), we can correct roughly $m^{r/2}$ errors. More generally, for any integer $r$ , our algorithm can correct any error pattern in $RM(m-(2r+2),m)$ , for which the same erasure pattern can be corrected in $RM(m-(r+1),m)$ . The results above are obtained by applying recent results of Abbe, Shpilka, and Wigderson (STOC, 2015) and Kudekar et al. (STOC, 2016) regarding the ability of RM codes to correct random erasures. The algorithm is based on solving a carefully defined set of linear equations and thus it is significantly different than other algorithms for decoding RM codes that are based on the recursive structure of the code. It can be seen as a more explicit proof of a result of Abbe et al. that shows a reduction from correcting erasures to correcting errors, and it also bares some similarities with the error-locating pair method of Pellikaan, Duursma, and Kotter that generalizes the Berlekamp–Welch algorithm for decoding Reed–Solomon codes.

Power Optimization in Random Wireless Networks

In this paper, we analyze the problem of power control in large, random wireless networks that are obtained by “erasing” a finite fraction of nodes from a regular $d$ -dimensional lattice of $N$ transmit-receive pairs. In this model, which has the important feature of a minimum distance between transmitter nodes, we find that when the network is infinite, power control is always feasible below a positive critical value of the users’ signal-to-interference-plus-noise ratio (SINR) target. Drawing on tools and ideas from statistical physics, we show how this problem can be mapped to the Anderson impurity model for diffusion in random media. In this way, by employing the so-called coherent potential approximation method, we calculate the average power in the system (and its variance) for 1-D and 2-D networks. This approach is equivalent to traditional techniques from random matrix theory and is in excellent agreement with the numerical simulations; however, it fails to predict when power control becomes infeasible. In this regard, even though infinitely large systems are always unstable beyond a critical value of the users’ SINR target, finite systems remain stable with high probability even beyond this critical SINR threshold. We calculate this probability by analyzing the density of low lying eigenvalues of an associated random Schrodinger operator, and we show that the network can exceed this critical SINR threshold by at least $ \mathop {{\mathcal O}}\nolimits ((\log N)^{-2/d})$ before undergoing a phase transition to the unstable regime. Finally, using the same techniques, we also calculate the tails of the distribution of transmit power in the system and the rate of convergence of the Foschini–Miljanic power control algorithm in the presence of random erasures.

Reed–Muller Codes for Random Erasures and Errors

This paper studies the parameters for which binary Reed-Muller (RM) codes can be decoded successfully on the binary erasure channel and binary symmetry channel, and, in particular, when can they achieve capacity for these two classical channels. Necessarily, this paper also studies the properties of evaluations of multivariate GF(2) polynomials on the random sets of inputs. For erasures, we prove that RM codes achieve capacity both for very high rate and very low rate regimes. For errors, we prove that RM codes achieve capacity for very low rate regimes, and for very high rates, we show that they can uniquely decode at about the square root of the number of errors at capacity. The proofs of these four results are based on different techniques, which we find interesting in their own right. In particular, we study the following questions about E(m, r), the matrix whose rows are the truth tables of all the monomials of degree ≤ r in m variables. What is the most (resp. least) number of random columns in E(m, r) that define a submatrix having full column rank (resp. full row rank) with high probability? We obtain tight bounds for very small (resp. very large) degrees r, which we use to show that RM codes achieve capacity for erasures in these regimes. Our decoding from random errors follows from the following novel reduction. For every linear code C of sufficiently high rate, we construct a new code C' obtained by tensorizing C, such that for every subset S of coordinates, if C can recover from erasures in S, then C' can recover from errors in S. Specializing this to the RM codes and using our results for erasures imply our result on the unique decoding of the RM codes at high rate. Finally, two of our capacity achieving results require tight bounds on the weight distribution of RM codes. We obtain such bounds extending the recent bounds from constant degree to linear degree polynomials.

Modeling Network Coded TCP: Analysis of Throughput and Energy Cost

We analyze the performance of TCP and TCP with network coding (TCP/NC) in lossy networks. We build upon the framework introduced by Padhye et al. and characterize the throughput behavior of classical TCP and TCP/NC as a function of erasure probability, round-trip time, maximum window size, and duration of the connection. Our analytical results show that network coding masks random erasures from TCP, thus preventing TCP’s performance degradation in lossy networks. It is further seen that TCP/NC has significant throughput gains over TCP. In addition, we study the cost of improving the goodput per user in a wireless network. We measure the cost in terms of number of base stations, which is highly correlated to the energy cost of a network provider. We show that increasing the available bandwidth may not necessarily lead to increase in goodput, particularly in lossy wireless networks using TCP. We show that using protocols such as TCP/NC, which are more resilient to erasures in the network, may lead to a goodput commensurate with the bandwidth dedicated to each user. By increasing goodput, users’ transactions are completed faster; thus, the resources dedicated to these users can be released to serve other requests, consequently reducing the cost for the network providers.

End-to-End Virtual MIMO Transmission in Ad Hoc Cognitive Radio Networks

Ad hoc cognitive radio networks (CRNs) have the potential for meeting the challenge of increasing radio spectrum efficiency. However, traditional control of ad hoc networking demands end-to-end information channel feedback, whose feasibility is hard in ad hoc CRNs due to the opportunistic nature of spectrum access. In this paper, we propose a virtual multiple-input multiple-output (MIMO) approach to facilitate error-resilient end-to-end transmission with no need for feedback information. An erasure-channel model is used to describe the randomness of outage caused by opportunistic links. At source node, a discrete Fourier transform (DFT)-based path-time code (PTC) is used to exploit path diversity. The a priori erasure probability is analyzed, and a pipeline scheduling scheme with unequal waiting periods is designed to reduce such probability. At destination node, knowledge of the a priori erasure probability is exploited to overcome the decoding challenge raised by the presence of random erasures. A joint sphere decoder (JSD) with a minimum mean-squared error sorted QR decomposition (MMSE-SQRD) effectively implements maximum a posteriori (MAP) probability decoding. This decoder simultaneously performs erasure identification and data decoding. Numerical results show that the proposed virtual MIMO framework can pave the way to efficient and reliable end-to-end transmission in ad hoc CRNs.

Unified Architecture for Reed-Solomon Decoder Combined With Burst-Error Correction

Reed-Solomon (RS) codes are widely used as forward correction codes (FEC) in digital communication and storage systems. Correcting random errors of RS codes have been extensively studied in both academia and industry. However, for burst-error correction, the research is still quite limited due to its ultra high computation complexity. In this brief, starting from a recent theoretical work, a low-complexity reformulated inversionless burst-error correcting (RiBC) algorithm is developed for practical applications. Then, based on the proposed algorithm, a unified VLSI architecture that is capable of correcting burst errors, as well as random errors and erasures, is firstly presented for multi-mode decoding requirements. This new architecture is denoted as unified hybrid decoding (UHD) architecture. It will be shown that, being the first RS decoder owning enhanced burst-error correcting capability, it can achieve significantly improved error correcting capability than traditional hard-decision decoding (HDD) design.

Robust network coding against path failures

In this study, an approach for robust network coding is introduced for multicast in a directed acyclic network in the presence of network edge failures. The proposed designs aim at combating the resulting path failures, which result in interestingly scalable solutions. A robust network coding scheme (RNC1) is proposed that, devising a rate-path diversity trade-off for the receivers, attains the post-failure capacity of the network with high probability. The scheme is receiver based and can also be applied for correcting random erasures. Next, a rate-guaranteed robust network coding scheme (RNC2) is proposed. The code guarantees the maximum rate for a predetermined number of path failures. The scheme, of course, attains the refined Singleton bound for the edge failure model. A path failure may not necessarily reduce the network capacity, as the remaining intact edges within the network may still facilitate backup paths from the source to the sinks. We introduce RNC3 to employ such backup paths in addition to the original paths and guarantee multicast at a certain rate in the presence of all edge/path failure patterns that do not reduce the capacity below this rate. All the three proposed schemes for multicast are robust to a number of edge failures that may, in general, exceed the refined Singleton bound. Our analyses indicate that the design complexities and the required field sizes grow as a function of the number of network paths, as opposed to the number of network edges because of prior schemes.

Burst erasure correcting LDPC codes

In this paper low-density parity-check (LDPC) codes are designed for burst erasure channels. Firstly, lower bounds for the maximum length erasure burst that can always be corrected with message-passing decoding are derived as a function of the parity-check matrix properties. We then show how parity-check matrices for burst erasure correcting LDPC codes can be constructed using superposition, where the burst erasure correcting performance of the resulting codes is derived as a property of the stopping set size of the base matrices and the choice of permutation matrices for the superposition. This result is then used to design both single burst erasure correcting LDPC codes which are also resilient to the presence of random erasures in the received bits and LDPC codes which can correct multiple erasure bursts in the same codeword.

A new LDPC decoding algorithm aided by segmented cyclic redundancy checks for magnetic recording channels

We introduce a new low-density parity-check (LDPC) decoding algorithm that exploits the cyclic redundancy check (CRC) information of data segments. By using the error detection property of the CRC, we can successively decode data segments of a codeword corrupted by random errors and erasures. The key idea is that the messages from the variable nodes with correct checksum are fixed to deterministic log likelihood ratio values during LDPC iterative decoding. This approach improves the decoding speed and codeword error rate without significant modification of the LDPC decoding structure. Moreover, the CRC is also used for an early stopping criterion of LDPC decoding. Simulation results verify our claims.

Burst Erasure Correction Codes With Low Decoding Delay

We present a new class of systematic, time-invariant, convolutional encoders suitable for low delay burst erasure correction. Specifically, we show that the new encoders have the shortest possible decoding delay required to correct all bursts of a given length with a fixed redundancy. By comparing the new encoders to maximum distance separable (MDS) codes, we show that the latter generally require either more redundancy or more delay to correct bursts of a given length. In addition, we show that the new encoders can achieve better performance than MDS codes on a simple two-state Markov erasure channel. Thus, we demonstrate the advantages of using cross packet coding for delay-sensitive applications such as Voice over Internet Protocol, video-conferencing, etc., on bursty packet networks. Finally, we discuss suitable performance measures for encoders designed to correct both burst and random erasures and report the results of a computer search for such hybrid encoders.

On binary majority-logic decodable codes (Corresp.)

Let V\prime be a binary (n,k) majority-logic decodable code with g\prime (X) as its generator polynomial and odd minimum distance d . Let V be the (n, k - 1) subset code generated by g\prime (X)(1 + X ). This correspondence shows that V is majority-logic deeodable with d + 1 orthogonal estimates. This fact is useful in the simultaneous correction of random errors and erasures.