Search

Generalized low-density parity-check (GLDPC) codes are a class of LDPC codes in which the standard single parity check (SPC) constraints are replaced by constraints defined by a linear block code. These stronger constraints typically result in improved error floor performance, due to better minimum distance and trapping set properties, at a cost of some increased decoding complexity. In this paper, we study spatially coupled generalized low-density parity-check (SC-GLDPC) codes and present a comprehensive analysis of these codes, including: (1) an iterative decoding threshold analysis of SC-GLDPC code ensembles demonstrating capacity approaching thresholds via the threshold saturation effect; (2) an asymptotic analysis of the minimum distance and free distance properties of SC-GLDPC code ensembles, demonstrating that the ensembles are asymptotically good; and (3) an analysis of the finite-length scaling behavior of both GLDPC block codes and SC-GLDPC codes based on a peeling decoder (PD) operating on a binary erasure channel (BEC). Results are compared to GLDPC block codes, and the advantages and disadvantages of SC-GLDPC codes are discussed. [ABSTRACT FROM AUTHOR]

Write-once memory (WOM) is a storage device consisting of binary cells that can only increase their levels. A $t$ -write WOM code is a coding scheme that makes it possible to write $t$ times to a WOM without decreasing the levels of any of the cells. The sum-rate of a WOM code is the ratio between the total number of bits written to the memory during the $t$ writes and the number of cells. It is known that the maximum possible sum-rate of a $t$ -write WOM code is $\log (t+1)$. This is also an achievable upper bound, both by information-theoretic arguments and through explicit constructions. While existing constructions of WOM codes are targeted at the sum-rate, we consider here two more figures of merit. The first one is the complexity of the encoding and decoding maps. The second figure of merit is the convergence rate, defined as the minimum code length $n(\delta)$ required to reach a point that is $\delta $ -close to the capacity region. One of our main results in this paper is a capacity-achieving construction of two-write WOM codes which has polynomial encoding/decoding complexity while the block length $n(\delta)$ required to be $\delta $ -close to capacity is significantly smaller than existing constructions. Using these two-write WOM codes, we then obtain three-write WOM codes that approach a sum-rate of 1.809 at relatively short block lengths. We also provide several explicit constructions of finite length three-write WOM codes; in particular, we achieve a sum-rate of 1.716 by using only 93 cells. Finally, we modify our two-write WOM codes to construct $\epsilon $ -error WOM codes of high rates and small probability of failure. [ABSTRACT FROM AUTHOR]

For any prime power $q$ , Mori and Tanaka introduced a family of $q$ -ary polar codes based on the $q\,\,\times \,\,q$ Reed–Solomon polarization kernels. For transmission over a $q$ -ary erasure channel, they also derived a closed-form recursion for the erasure probability of each effective channel. In this paper, we use that expression to analyze the finite-length scaling of these codes on the $q$ -ary erasure channel with erasure probability $\epsilon \in (0,1)$. Our primary result is that for any $\gamma >0$ and $\delta >0$ , there is a $q_{0}$ , such that for all $q\geq q_{0}$ , the fraction of effective channels with erasure rate at most $N^{-\gamma }$ is at least $1-\epsilon -O(N^{-1/2+\delta })$ , where $N=q^{n}$ is the blocklength. Since this fraction cannot be larger than $1-\epsilon -O(N^{-1/2})$ , this establishes near-optimal finite-length scaling for this family of codes. Our approach can be seen as an extension of a similar analysis for binary polar codes by Hassani, Alishahi, and Urbanke. A similar analysis is also considered for $q$ -ary polar codes with $m \times m$ polarizing matrices. This separates the effect of the alphabet size from the effect of the matrix size. If the polarizing matrix at each stage is drawn independently and uniformly from the set of invertible $m \times m$ matrices, then the linear operator associated with the Lyapunov function analysis can be written in the closed form. To prove near-optimal scaling for polar codes with fixed $q$ as $m$ increases, however, two technical obstacles remain. Thus, we conclude by stating two concrete mathematical conjectures that, if proven, would imply near-optimal scaling for fixed $q$. [ABSTRACT FROM AUTHOR]

Regular spatially-coupled low-density parity-check ensembles have gained significant interest, since they were shown to universally achieve the capacity of binary memoryless channels under low-complexity belief-propagation decoding. In this paper, we focus primarily on the performance of these ensembles over binary channels affected by bursts of erasures. We first develop an analysis of the finite length performance for a single burst per code word and no errors otherwise. We first assume that the burst erases a complete spatial position, modeling for instance node failures in distributed storage. We provide new tight lower bounds for the block erasure probability ( P_ \mathrm \scriptscriptstyle B ) at finite block length and bounds on the coupling parameter for being asymptotically able to recover the burst. We further show that expurgating the ensemble can improve the block erasure probability by several orders of magnitude. Later we extend our methodology to more general channel models. In a first extension, we consider bursts that can start at a random location in the code word and span across multiple spatial positions. Besides the finite length analysis, we determine by means of density evolution the maximum correctable burst length. In a second extension, we consider the case where in addition to a single burst, random bit erasures may occur. Finally, we consider a block erasure channel model which erases each spatial position independently with some probability $p$ , potentially introducing multiple bursts simultaneously. All results are verified using Monte-Carlo simulations. [ABSTRACT FROM PUBLISHER]

BATS codes were proposed for communication through networks with packet loss. A BATS code consists of an outer code and an inner code. The outer code is a matrix generation of a fountain code, which works with the inner code that comprises random linear coding at the intermediate network nodes. In this paper, the performance of finite-length BATS codes is analyzed with respect to both belief propagation (BP) decoding and inactivation decoding. Our results enable us to evaluate efficiently the finite-length performance in terms of the number of batches used for decoding ranging from 1 to a given maximum number, and provide new insights on the decoding performance. Specifically, for a fixed number of input symbols and a range of the number of batches used for decoding, we obtain recursive formulae to calculate the stopping time distribution of BP decoding and the inactivation probability in inactivation decoding. We also find that both the failure probability of BP decoding and the expected number of inactivations in inactivation decoding can be expressed in a power-sum form where the number of batches appears only as the exponent. This power-sum expression reveals clearly how the decoding failure probability and the expected number of inactivation decrease with the number of batches. When the number of batches used for decoding follows a Poisson distribution, we further derive recursive formulas with potentially lower computational complexity for both decoding algorithms. For the BP decoder that consumes batches one by one, three formulae are provided to characterize the expected number of consumed batches until all the input symbols are decoded. [ABSTRACT FROM PUBLISHER]

Index coding models broadcast networks in which a sender sends different messages to different receivers simultaneously, where each receiver may know some of the messages a priori. The aim is to find the minimum (normalized) index codelength that the sender sends. This paper considers unicast index coding, where each receiver requests exactly one message, and each message is requested by exactly one receiver. Each unicast index-coding instances can be fully described by a directed graph and vice versa, where each vertex corresponds to one receiver. For any directed graph representing a unicast index-coding instance, we show that if a maximum acyclic induced subgraph (MAIS) is obtained by removing two or fewer vertices from the graph, then the minimum index codelength equals the number of vertices in the MAIS, and linear codes are optimal for the corresponding index-coding instance. Using this result, we solved all unicast index-coding instances with up to five receivers, which correspond to all graphs with up to five vertices. For 9818 non-isomorphic graphs among all graphs up to five vertices, we obtained the minimum index codelength for all message alphabet sizes; for the remaining 28 graphs, we obtained the minimum index codelength if the message alphabet size is k^2 for any positive integer $k$ . This work complements the result by Arbabjolfaei et al. (ISIT 2013), who solved all unicast index-coding instances with up to five receivers in the asymptotic regime, where the message alphabet size tends to infinity. [ABSTRACT FROM PUBLISHER]

The polarization process of polar codes over a prime $q$ -ary alphabet is studied. Recently, it has been shown that the blocklength of polar codes with prime alphabet size scales polynomially with respect to the inverse of the gap between code rate and channel capacity. However, except for the binary case, the degree of the polynomial in the bound is extremely large. In this paper, a different approach to computing the degree of this polynomial for any prime alphabet size is shown. This approach yields a lower degree polynomial for various values of the alphabet size that were examined. It is also shown that even lower degree polynomial can be computed with an additional numerical effort. [ABSTRACT FROM AUTHOR]

We study a noiseless broadcast link serving K users whose requests arise from a library of N files. Every user is equipped with a cache of size M files each. It has been shown that by splitting all the files into packets and placing individual packets in a random independent manner across all the caches prior to any transmission, at most N/M file transmissions are required for any set of demands from the library. The achievable delivery scheme involves linearly combining packets of different files following a greedy clique cover solution to the underlying index coding problem. This remarkable multiplicative gain of random placement and coded delivery has been established in the asymptotic regime when the number of packets per file F scales to infinity. The asymptotic coding gain obtained is roughly t=KM/N . In this paper, we initiate the finite-length analysis of random caching schemes when the number of packets F is a function of the system parameters . Furthermore, for any clique cover-based coded delivery and a large class of random placement schemes that include the existing ones, we show that the number of packets required to get a multiplicative gain of ({4}/{3})g is at least O(({g}/{K})(N/M)^{g-1})$ . We design a new random placement and an efficient clique cover-based delivery scheme that achieves this lower bound approximately. We also provide tight concentration results that show that the average (over the random placement involved) number of transmissions concentrates very well requiring only a polynomial number of packets in the rest of the system parameters. [ABSTRACT FROM AUTHOR]

Spatially-coupled low-density parity-check (SC-LDPC) codes are known to have excellent asymptotic properties. Much less is known regarding their finite-length performance. We propose a scaling law to predict the error probability of finite-length spatially coupled code ensembles when transmission takes place over the binary erasure channel. We discuss how the parameters of the scaling law are connected to fundamental quantities appearing in the asymptotic analysis of these ensembles and we verify that the predictions of the scaling law fit well to the data derived from simulations over a wide range of parameters. The ultimate goal of this line of research is to develop analytic tools for the design of SC-LDPC codes under practical constraints. [ABSTRACT FROM AUTHOR]

This paper considers the performance, on the binary erasure channel, of low-density parity-check (LDPC) codes without repeated edges in their Tanner graphs. A modification to existing finite-length analysis algorithms is presented for these codes. [ABSTRACT FROM AUTHOR]

In this paper, we study linear encoding for a pair of correlated Gaussian sources transmitted over a two-user Gaussian broadcast channel in the presence of unit-delay noiseless feedback, abbreviated as the GBCF. Each pair of source samples is transmitted using a linear transmission scheme in a finite number of channel uses. We investigate three linear transmission schemes: A scheme based on the Ozarow–Leung (OL) code, a scheme based on the linear quadratic Gaussian (LQG) code of Ardestanizadeh et al., and a novel scheme derived in this paper using a dynamic programming (DP) approach. For the OL and LQG schemes we present lower and upper bounds on the minimal number of channel uses needed to achieve a target mean-square error (MSE) pair. For the LQG scheme in the symmetric setting, we identify the optimal scaling of the sources, which results in a significant improvement of its finite horizon performance, and, in addition, characterize the (exact) minimal number of channel uses required to achieve a target MSE. Finally, for the symmetric setting, we show that for any fixed and finite number of channel uses, the DP scheme achieves an MSE lower than the MSE achieved by either the LQG or the OL schemes. [ABSTRACT FROM AUTHOR]

Consider a binary-input memoryless output-symmetric channel \(W\) . Such a channel has a capacity, call it \(I(W)\) , and for any \(R0\) , then the required block-length \(N\) scales in terms of the rate \(R < I(W)\) as \(N \geq {\alpha }/{(I(W)-R)^{\underline {\mu }}}\) , where \(\alpha \) is a positive constant that depends on \(P_{\rm e}\) and \(I(W)\) . We show that \(\underline {\mu } = 3.579\) is a valid choice, and we conjecture that indeed the value of \(\underline {\mu }\) can be improved to \(\underline {\mu }=3.627\) , the parameter for the binary erasure channel. Also, we show that with the same requirement on the sum of Bhattacharyya parameters, the block-length scales in terms of the rate like \(N \leq {\beta }/{(I(W)-R)^{\overline {\mu }}}\) , where \(\beta \) is a constant that depends on \(P_{\rm e}\) and \(I(W)\) , and \(\overline {\mu }=6\) . [ABSTRACT FROM PUBLISHER]

We investigate the behavior of iteratively decoded low-density parity-check (LDPC) codes over the binary erasure channel in the so-called "waterfall region." We show that the performance curves in this region follow a simple scaling law. We conjecture that essentially the same scaling behavior applies in a much more general setting and we provide some empirical evidence to support this conjecture. The scaling law, together with the error floor expressions developed previously, can be used for a fast finite-length optimization. [ABSTRACT FROM AUTHOR]

Consider the case where consecutive blocks of N letters of a semi-infinite individual sequence X over a finite alphabet are being compressed into binary sequences by some one-to-one mapping. No a priori information about X is available at the encoder, which must therefore adopt a universal data-compression algorithm. It is known that there exist a number of asymptotically optimal universal data compression algorithms (e.g., the Lempel—Ziv (LZ) algorithm, Context tree algorithm and an adaptive Hufmann algorithm) such that when successively applied to N-blocks then, the best error-free compression for the particular individual sequence X is achieved as N tends to infinity. The best possible compression that may be achieved by any universal data compression algorithm for finite N-blocks is discussed. Essential optimality for the compression of finite-length sequences is defined. It is shown that the LZ77 universal compression of N-blocks is essentially optimal for finite N-blocks. Previously, it has been demonstrated that a universal context tree compression of N blocks is essentially optimal as well. [ABSTRACT FROM AUTHOR]

This paper derives an improved sphere-packing (ISP) bound for finite-length error-correcting codes whose transmission takes place over symmetric memoryless channels, and the codes are decoded with an arbitrary list decoder. We first review classical results, i.e., the 1959 sphere-packing (SP59) bound of Shannon for the Gaussian channel, and the 1967 sphere-packing (SP67) bound of Shannon et al. for discrete memoryless channels. An improvement on the SP67 bound, as suggested by Valembois and Fossorier, is also discussed. These concepts are used for the derivation of a new lower bound on the error probability of list decoding (referred to as the ISP bound) which is uniformly tighter than the SP67 bound and its improved version. The ISP bound is applicable to symmetric memoryless channels, and some of its applications are presented. Its tightness under maximum-likelihood (ML) decoding is studied by comparing the ISP bound to previously reported upper and lower bounds on the ML decoding error probability, and also to computer simulations of iteratively decoded turbo-like codes. This paper also presents a technique which performs the entire calculation of the SP59 bound in the logarithmic domain, thus facilitating the exact calculation of this bound for moderate to large block lengths without the need for the asymptotic approximations provided by Shannon. [ABSTRACT FROM AUTHOR]

This paper is devoted to the finite-length analysis of turbo decoding over the binary erasure channel (BEC). The performance of iterative belief-propagation decoding of low-density parity-check (LDPC) codes over the BEC can be characterized in terms of stopping sets. We describe turbo decoding on the BEC which is simpler than turbo decoding on other channels. We then adapt the concept of stopping sets to turbo decoding and state an exact condition for decoding failure. Apply turbo decoding until the transmitted codeword has been recovered, or the decoder fails to progress further. Then the set of erased positions that will remain when the decoder stops is equal to the unique maximum-size turbo stopping set which is also a subset of the set of erased positions. Furthermore, we present some improvements of the basic turbo decoding algorithm on the BEC. The proposed improved turbo decoding algorithm has substantially better error performance as illustrated by the given simulation results. Finally, we give an expression for the turbo stopping set size enumerating function under the uniform interleaver assumption, and an efficient enumeration algorithm of small-size turbo stopping sets for a particular interleaver. The solution is based on the algorithm proposed by Garello et al. in 2001 to compute an exhaustive list of all low-weight codewords in a turbo code. [ABSTRACT FROM AUTHOR]

Improved upper bounds on the blocklength required to communicate over binary-input channels using polar codes, below some given error probability, are derived. For that purpose, an improved bound on the number of non-polarizing channels is obtained. The main result is that the blocklength required to communicate reliably scales at most as \(O((I(W)-R)^{-5.702})\) , where \(R\) is the code rate and \(I(W)\) is the symmetric capacity of the channel \(W\) . The results are then extended to polar lossy source coding at rate \(R\) of a source with symmetric distortion-rate function \(D(\cdot )\) . The blocklength required scales at most as \(O((D^{0})^{-5.702})\) , where \(D^{0}\) is the maximal allowed gap between the actual average (or typical) distortion and \(D(R)\) . [ABSTRACT FROM PUBLISHER]

Presents a study that analyzed the finite-length of low-density parity-check (LDPC) codes when used over the binary erasure channel. Background on LDPC codes under belief propagation decoding; Comparison of LDPC ensembles between iterative decoding and maximum-likelihood decoding; Information on the characterization of the average bit and block erasure probabilities for a given regular ensemble of LDPC codes.

The nonlinear complexity of binary sequences and its connections with Lempel-Ziv complexity is studied in this paper. A new recursive algorithm is presented, which produces the minimal nonlinear feedback shift register of a given binary sequence. Moreover, it is shown that the eigenvalue profile of a sequence uniquely determines its nonlinear complexity profile, thus establishing a connection between Lempel-Ziv complexity and nonlinear complexity. Furthermore, a lower bound for the Lempel-Ziv compression ratio of a given sequence is proved that depends on its nonlinear complexity. [ABSTRACT FROM AUTHOR]

In this correspondence, a generalization of rateless codes is proposed. The proposed codes provide unequal error protection (UEP). The asymptotic properties of these codes under the iterative decoding are investigated. Moreover, upper and lower bounds on maximum-likelihood (ML) decoding error probabilities of finite-length LT and Rapt or codes for both equal and unequal error protection schemes are derived. Further, our work is verified with simulations. Simulation results indicate that the pro. posed codes provide desirable UEP. We also note that the UEP property does not impose a considerable drawback on the overall performance of the codes. Moreover, we discuss that the proposed codes can provide unequal recovery time (URT). This means that given a target bit error rate, different parts of information bits can be decoded after receiving different amounts of encoded bits. This implies that the information bits can be recovered in a progressive manner. This URT property may be used for sequential data recovery in video/audio streaming. [ABSTRACT FROM AUTHOR]

In this correspondence, blind channel estimators exploiting finite alphabet constraints are discussed for orthogonal frequency-division multiplexing (OFDM) systems. Considering the channel and data jointly, a joint maximum-likelihood (JML) algorithm is described, along with identifiability conditions in the noise-free case. This approach enables development of general identifiability conditions for the minimum-distance (MD) finite alphabet blind algorithm of Zhou and Giannakis. Both the JML and MD algorithms suffer from high numerical complexity, as they rely on exhaustive search methods to resolve a large number of ambiguities. We present a substantially more efficient blind algorithm, the reduced complexity minimum distance (RMD) algorithm, by exploiting properties of the assumed finite-length impulse response (FIR) channel. The RMD algorithm exploits constraints on the unwrapped phase of FIR systems and results in significant reductions in numerical complexity over existing methods. In many cases, the RMD approach is able to completely eliminate the exhaustive search of the JML and MD approaches, while providing channel estimates of the same quality. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new channel model termed as the $q$ -ary multi-bit channel. This channel models a memory device, where $q$ -ary symbols ($q=2^{s}$) are stored in the form of current/voltage levels. The symbols are read in a measurement process, which provides a symbol bit in each measurement step, starting from the most significant bit. An error event occurs when not all the symbol bits are known. To deal with such error events, we use GF($q$) low-density parity-check (LDPC) codes and analyze their decoding performance. We start with iterative-decoding threshold analysis and derive optimal edge-label distributions for maximizing the decoding threshold. We later move to a finite-length iterative-decoding analysis and propose an edge-labeling algorithm for the improved decoding performance. We then provide a finite-length maximum-likelihood decoding analysis for both the standard non-binary random ensemble and LDPC ensembles. Finally, we demonstrate by simulations that the proposed edge-labeling algorithm improves the finite-length decoding performance by orders of magnitude. [ABSTRACT FROM AUTHOR]

Due to its high data density and longevity, DNA is considered a promising medium for satisfying ever-increasing data storage needs. However, the diversity of errors that occur in DNA sequences makes efficient error-correction a challenging task. This paper aims to address simultaneously correcting two types of errors, namely, short tandem duplication and edit errors, where an edit error may be a substitution, deletion, or insertion. We focus on tandem repeats of length at most 3 and design codes for correcting an arbitrary number of duplication errors and one edit error. Because an edited symbol can be duplicated many times (as part of substrings of various lengths), a single edit can affect an unbounded substring of the retrieved word. However, we show that with appropriate preprocessing, the effect may be limited to a substring of finite length, thus making efficient error-correction possible. We construct a code for correcting the aforementioned errors and provide lower bounds for its rate. Compared to optimal codes correcting only duplication errors, numerical results show that the asymptotic cost of protecting against an additional edit is only 0.003 bits/symbol when the alphabet has size 4, an important case corresponding to data storage in DNA. [ABSTRACT FROM AUTHOR]

Shannon in his 1956 seminal paper introduced the concept of the zero error capacity, C0 , of a noisy channel. This is defined as the least upper bound of rates, at which, it is possible to transmit information with zero probability of error. At present not many codes are known to achieve the zero error capacity. In this paper, some codes which achieve zero error capacities in limited magnitude error channels are described. The code lengths of these zero error capacity achieving codes can be of any finite length $n=1,2,\ldots$ , in contrast to the long lengths required for the known regular capacity achieving codes, such as turbo codes, LDPC codes, and polar codes. Both wrap around and non-wrap around limited magnitude error models are considered in this paper. For non-wrap around error model, the exact value of zero error capacities is derived, and optimal non-systematic and systematic codes are designed. The non-systematic codes achieve the zero error capacity with any finite length. The optimal systematic codes achieve the systematic zero error capacity of the channel, which is defined as the zero error capacity with the additional requirements that the communication must be carried out with a systematic code. It is also shown that the rates of the proposed systematic codes are equal to or approximately equal to the zero error capacity of the channel. For the wrap around model bounds are derived for the zero error capacity and in many cases the bounds give the exact value. In addition, optimal wrap around non-systematic and systematic codes are developed which either achieve or are close to achieving the zero error capacity with finite length. [ABSTRACT FROM PUBLISHER]

We present the tree-structure expectation propagation (Tree-EP) algorithm to decode low-density parity-check (LDPC) codes over discrete memoryless channels (DMCs). Expectation propagation generalizes belief propagation (BP) in two ways. First, it can be used with any exponential family distribution over the cliques in the graph. Second, it can impose additional constraints on the marginal distributions. We use this second property to impose pairwise marginal constraints over pairs of variables connected to a check node of the LDPC code's Tanner graph. Thanks to these additional constraints, the Tree-EP marginal estimates for each variable in the graph are more accurate than those provided by BP. We also reformulate the Tree-EP algorithm for the binary erasure channel (BEC) as a peeling-type algorithm (TEP) and we show that the algorithm has the same computational complexity as BP and it decodes a higher fraction of errors. We describe the TEP decoding process by a set of differential equations that represents the expected residual graph evolution as a function of the code parameters. The solution of these equations is used to predict the TEP decoder performance in both the asymptotic regime and the finite-length regimes over the BEC. While the asymptotic threshold of the TEP decoder is the same as the BP decoder for regular and optimized codes, we propose a scaling law for finite-length LDPC codes, which accurately approximates the TEP improved performance and facilitates its optimization. [ABSTRACT FROM PUBLISHER]

Sparse superposition codes, also referred to as sparse regression codes (SPARCs), are a class of codes for efficient communication over the AWGN channel at rates approaching the channel capacity. In a standard SPARC, codewords are sparse linear combinations of columns of an i.i.d. Gaussian design matrix, while in a spatially coupled SPARC the design matrix has a block-wise structure, where the variance of the Gaussian entries can be varied across blocks. A well-designed spatial coupling structure can significantly enhance the error performance of iterative decoding algorithms such as Approximate Message Passing (AMP). In this paper, we obtain a non-asymptotic bound on the probability of error of spatially coupled SPARCs with AMP decoding. Applying this bound to a simple band-diagonal design matrix, we prove that spatially coupled SPARCs with AMP decoding achieve the capacity of the AWGN channel. The bound also highlights how the decay of error probability depends on each design parameter of the spatially coupled SPARC. An attractive feature of AMP decoding is that its asymptotic mean squared error (MSE) can be predicted via a deterministic recursion called state evolution. Our result provides the first proof that the MSE concentrates on the state evolution prediction for spatially coupled designs. Combined with the state evolution prediction, this result implies that spatially coupled SPARCs with the proposed band-diagonal design are capacity-achieving. Using the proof technique used to establish the main result, we also obtain a concentration inequality for the MSE of AMP applied to compressed sensing with spatially coupled design matrices. Finally, we provide numerical simulation results that demonstrate the finite length error performance of spatially coupled SPARCs. The performance is compared with coded modulation schemes that use LDPC codes from the DVB-S2 standard. [ABSTRACT FROM AUTHOR]

In this paper, we propose a systematic low density generator matrix (LDGM) code ensemble, which is defined by the Bernoulli process. We prove that, under maximum likelihood (ML) decoding, the proposed ensemble can achieve the capacity of binary-input output symmetric (BIOS) memoryless channels in terms of bit error rate (BER). The proof technique reveals a new mechanism, different from lowering down frame error rate (FER), that the BER can be lowered down by assigning light codeword vectors to light information vectors. The finite length performance is analyzed by deriving an upper bound and a lower bound, both of which are shown to be tight in the high signal-to-noise ratio (SNR) region. To improve the waterfall performance, we construct the systematic convolutional LDGM (SysConv-LDGM) codes by a random splitting process. The SysConv-LDGM codes are easily configurable in the sense that any rational code rate can be realized without complex optimization. As a universal construction, the main advantage of the SysConv-LDGM codes is their near-capacity performance in the waterfall region and predictable performance in the error-floor region that can be lowered down to any target as required by increasing the density of the uncoupled LDGM codes. Numerical results are also provided to verify our analysis. [ABSTRACT FROM AUTHOR]

In this paper, we analyze classical data compression with quantum side information (also known as the classical-quantum Slepian–Wolf protocol) in the so-called large and moderate deviation regimes. In the non-asymptotic setting, the protocol involves compressing classical sequences of finite length $n$ and decoding them with the assistance of quantum side information. In the large deviation regime, the compression rate is fixed, and we obtain bounds on the error exponent function, which characterizes the minimal probability of error as a function of the rate. Devetak and Winter showed that the asymptotic data compression limit for this protocol is given by a conditional entropy. For any protocol with a rate below this quantity, the probability of error converges to one asymptotically and its speed of convergence is given by the strong converse exponent function. We obtain finite blocklength bounds on this function, and determine exactly its asymptotic value. In the moderate deviation regime for the compression rate, the latter is no longer considered to be fixed. It is allowed to depend on the blocklength $n$ , but assumed to decay slowly to the asymptotic data compression limit. Starting from a rate above this limit, we determine the speed of convergence of the error probability to zero and show that it is given in terms of the conditional information variance. [ABSTRACT FROM AUTHOR]

A locally recoverable code (LRC code) is a code over a finite alphabet, such that every symbol in the encoding is a function of a small number of other symbols that form a recovering set. In this paper, we derive new finite-length and asymptotic bounds on the parameters of LRC codes. For LRC codes with a single recovering set for every coordinate, we derive an asymptotic Gilbert–Varshamov type bound for LRC codes and find the maximum attainable relative distance of asymptotically good LRC codes. Similar results are established for LRC codes with two disjoint recovering sets for every coordinate. For the case of multiple recovering sets (the availability problem), we derive a lower bound on the parameters using expander graph arguments. Finally, we also derive finite-length upper bounds on the rate and the distance of LRC codes with multiple recovering sets. [ABSTRACT FROM AUTHOR]

The variable-length source coding problem allowing the error probability up to some constant is considered for general sources. In this problem, the optimum mean codeword length of variable-length codes has already been determined. On the other hand, in this paper, we focus on the overflow (or excess codeword length) probability instead of the mean codeword length. The infimum of overflow thresholds under the constraint that both of the error probability and the overflow probability are smaller than or equal to some constant is called the optimum overflow threshold. In this paper, we first derive finite-length upper and lower bounds on these probabilities so as to analyze the optimum overflow thresholds. Then, by using these bounds, we determine the general formula of the optimum overflow thresholds in both of the first-order and second-order forms. Next, we consider another expression of the derived general formula so as to reveal the relationship with the optimum coding rate in the fixed-length source coding problem. Finally, we apply the general formula derived in this paper to the case of stationary memoryless sources. [ABSTRACT FROM AUTHOR]

This paper provides a comprehensive analysis of nonbinary low-density parity check (LDPC) codes built out of protographs. We consider both random and constrained edge-weight labeling, and refer to the former as the unconstrained nonbinary protograph-based LDPC codes (U-NBPB codes) and to the latter as the constrained nonbinary protograph-based LDPC codes (C-NBPB codes). Equipped with combinatorial definitions extended to the nonbinary domain, ensemble enumerators of codewords, trapping sets, stopping sets, and pseudocodewords are calculated. The exact enumerators are presented in the finite-length regime, and the corresponding growth rates are calculated in the asymptotic regime. An EXIT chart tool for computing the iterative decoding thresholds of protograph-based LDPC codes is presented, followed by several examples of finite-length U-NBPB and C-NBPB codes with high performance. Throughout this paper, we provide accompanying examples, which demonstrate the advantage of nonbinary protograph-based LDPC codes over their binary counterparts and over random constructions. The results presented in this paper advance the analytical toolbox of nonbinary graph-based codes. [ABSTRACT FROM AUTHOR]

The performance of low-density parity-check (LDPC) codes in the error floor region is closely related to some substructures of the code’s Tanner graph, collectively referred to as trapping sets (TSs). In this paper, we study the asymptotic average number of different types of trapping sets such as elementary TSs (ETS), leafless ETSs (LETS), absorbing sets (ABS), elementary ABSs (EABS), and stopping sets (SS), in random variable-regular and irregular LDPC code ensembles. We demonstrate that, regardless of the type of the TS, as the code’s length tends to infinity, the average number of a given structure tends to infinity, to a positive constant, or to zero, if the structure contains no cycle, only one cycle, or more than one cycle, respectively. For the case where the structure contains a single cycle, we derive the asymptotic expected multiplicity of the structure by counting the average number of its constituent cycles and all the possible ways that the structure can be constructed from the cycle. This, in general, involves computing the expected number of cycles of a certain length with a certain given combination of node degrees, or computing the expected number of cycles of a certain length expanded to the desired structure by the connection of trees to its nodes. The asymptotic results obtained in this work, which are independent of the block length and only depend on the code’s degree distributions, are shown to be accurate even for finite-length codes. [ABSTRACT FROM AUTHOR]

A novel code construction based on spatially coupled low-density parity-check (SC-LDPC) codes is presented. The proposed code ensembles are comprised of several protograph-based chains characterizing individual SC-LDPC codes. We demonstrate that the code ensembles obtained by connecting appropriately chosen individual SC-LDPC code chains at specific points have improved iterative decoding thresholds. In addition, the connected chain ensembles have a smaller decoding complexity required to achieve a specific bit error probability compared to the individual code chains. Moreover, we demonstrate that, like the individual component chains, the proposed constructions have a typical minimum distance that grows linearly with block length. Finally, we show that the improved asymptotic properties of the connected chain ensembles also translate into improved finite length performance. [ABSTRACT FROM AUTHOR]

We establish a general formula for the maximum size of finite length block codes with minimum pairwise distance no less than $d$. The achievability argument involves an iterative construction of a set of radius- $d$ balls, each centered at a codeword. We demonstrate that the number of such balls that cover the entire code space cannot exceed this maximum size. Our approach can be applied to codes $i$) with elements over arbitrary code alphabets, and $ii$) under a broad class of distance measures. Our formula indicates that the maximum code size can be fully characterized by the cumulative distribution function of the distance measure evaluated at two independent and identically distributed random codewords. When the two random codewords assume a uniform distribution over the entire code alphabet, our formula recovers and thus naturally generalizes the Gilbert-Varshamov (GV) lower bound. Finally, we extend our study to the asymptotic setting. [ABSTRACT FROM AUTHOR]

Private information retrieval (PIR) is the problem of retrieving, as efficiently as possible, one out of $K$ messages from $N$ non-communicating replicated databases (each holds all $K$ messages) while keeping the identity of the desired message index a secret from each individual database. Symmetric PIR (SPIR) is a generalization of PIR to include the requirement that beyond the desired message, the user learns nothing about the other $K-1$ messages. The information theoretic capacity of SPIR (equivalently, the reciprocal of minimum download cost) is the maximum number of bits of desired information that can be privately retrieved per bit of downloaded information. We show that the capacity of SPIR is $1-1/N$ regardless of the number of messages $K$ , if the databases have access to common randomness (not available to the user) that is independent of the messages, in the amount that is at least $1/(N-1)$ bits per desired message bit. Otherwise, if the amount of common randomness is less than $1/(N-1)$ bits per message bit, then the capacity of SPIR is zero. Extensions to the capacity region of SPIR and the capacity of finite length SPIR are provided. [ABSTRACT FROM AUTHOR]

In this paper, the second moment of the ripple size during the Luby transform (LT) decoding process is analyzed. Combined with a result by Karp et al. (2004) stating that the expectation of the ripple size is of the order of k, our study gives bounds on the error probability of the LT decoder. Furthermore, an analytic expression for the variance of the ripple size up to terms of constant order is given, and the expression of Karp et al. for the expectation of the ripple size is refined up to terms of the order of 1/k. This provides a first step toward an analytic finite-length analysis of LT decoding. [ABSTRACT FROM PUBLISHER]

Protograph-based LDPC and generalized LDPC (G-LDPC) codes have the advantages of a simple design procedure and highly structured encoders and decoders. The design of such “protograph-based codes” relies on what is effectively a computer-based search. As such, following Gallager, it is prudent to restrict the search to a “good ensemble,” for example, an ensemble whose minimum distance grows linearly with codeword length. A good ensemble can also mean one with good stopping set, trapping set, or pseudocodeword properties. In this paper, ensemble codeword weight enumerators for finite-length LDPC and G-LDPC codes based on protographs were derived, and then the asymptotic case was considered. The asymptotic results allow us to determine whether or not the typical relative minimum distance in the ensemble grows linearly with codeword length. Then, the codeword weight enumerator technique is adapted to yield ensemble stopping set, trapping set, and pseudocodeword enumerators for protograph LDPC and G-LDPC codes. In this case, the asymptotic results allow us to determine whether or not the typical relative smallest stopping set size, trapping set size, and pseudoweight grows linearly with codeword length. Trapping set enumerators for G-LDPC code ensembles represent a more complex problem which we do not consider here. [ABSTRACT FROM AUTHOR]

It is proven in this work that it is NP-complete to exhaustively enumerate small error-prone substructures in arbitrary, finite-length low-density parity-check (LDPC) codes. Two error-prone patterns of interest include stopping sets for binary erasure channels (BECs) and trapping sets for general memoryless symmetric channels. Despite the provable hardness of the problem, this work provides an exhaustive enumeration algorithm that is computationally affordable when applied to codes of practical short lengths n ≈ 500. By exploiting the sparse connectivity of LDPC codes, the stopping sets of size ≤ 13 and the trapping sets of size ≤ 11 can be exhaustively enumerated. The central theorem behind the proposed algorithm is a new provably tight upper bound on the error rates of iterative decoding over BECs. Based on a tree-pruning technique, this upper bound can be iteratively sharpened until its asymptotic order equals that of the error floor. This feature distinguishes the proposed algorithm from existing non-exhaustive ones that correspond to finding lower bounds of the error floor. The upper bound also provides a worst case performance guarantee that is crucial to optimizing LDPC codes when the target error rate is beyond the reach of Monte Carlo simulation. Numerical experiments on both randomly and algebraically constructed LDPC codes demonstrate the efficiency of the search algorithm and its significant value for finite-length code optimization. [ABSTRACT FROM AUTHOR]

This paper describes a method for generating long sample paths of accurate fractional Gaussian noise (fGn), the increment process of fractional Brownian motion (fBm). The method is based on a Wold decomposition in which fGn is expressed as the output of a finite impulse response filter with discrete white Gaussian noise as input. The form of the ideal filter is derived analytically in the continuous-time case. For the finite-length discrete-time case, an iterative projection algorithm incorporating a Newton-Raphson step is described for computing the coefficients of a length-N filter in a time approximately proportional to N. Fast convolution of discrete white Gaussian noise with the computed filter impulse response yields arbitrarily long sequences which exactly match the correlation structure of fGn over a finite range of lags. For values of the Hurst parameter H smaller than a critical value Hcrit ≈ 0.85, and large N, the finite-length autocorrelation sequence of fGn is positive definite and this range of lags can be as large as the filter autocorrelation length. When H > Hcrit, the finite-length autocorrelation sequence of fGn is no longer positive definite and a modification is made to allow a Wold decomposition. The generated sequences then exactly match the autocorrelation structure of fGn over a more restricted range of lags which becomes smaller as H approaches unity. [ABSTRACT FROM AUTHOR]

The problem of finding the length of a shortest feedback shift register that generates a given finite-length sequence is considered. An efficient algorithm for the determination of' the span is proposed, that takes advantage of the special block structure of the associated system of linear equations. The span distribution of finite-length binary sequences is also studied. [ABSTRACT FROM AUTHOR]

A counter braid (CB) is a novel counter architecture introduced by Lu et al. in 2007 for per-flow measurements on high-speed links which can be decoded with low complexity using message passing (MP). CBs achieve an asymptotic compression rate (under optimal decoding) that matches the entropy lower bound of the flow size distribution. In this paper, we apply the concept of spatial coupling to CBs to improve the performance of the original CBs and analyze the performance of the resulting spatially-coupled CBs (SC-CBs). We introduce an equivalent bipartite graph representation of CBs with identical iteration-by-iteration finite-length and asymptotic performance. Based on this equivalent representation, we then analyze the asymptotic performance of single-layer CBs and SC-CBs under the MP decoding algorithm proposed by Lu et al.. In particular, we derive the potential threshold of the uncoupled system and show that it is equal to the area threshold. We also derive the Maxwell decoder for CBs and prove that the potential threshold is an upper bound on the Maxwell decoding threshold, which, in turn, is a lower bound on the maximum a posteriori (MAP) decoding threshold. We then show that the area under the extended MP extrinsic information transfer curve (defined for the equivalent graph), computed for the expected residual CB graph when a peeling decoder equivalent to the MP decoder stops, is equal to zero precisely at the area threshold. This, combined with the analysis of the Maxwell decoder and simulation results, leads us to the conjecture that the potential threshold is, in fact, equal to the Maxwell decoding threshold and hence a lower bound on the MAP decoding threshold. Interestingly, SC-CBs do not show the well-known phenomenon of threshold saturation of the MP decoding threshold to the potential threshold characteristic of spatially-coupled low-density parity-check codes and other coupled systems. However, SC-CBs yield better MP decoding thresholds than their uncoupled counterparts. Finally, we also consider SC-CBs as a compressed sensing scheme and show that low undersampling factors can be achieved. [ABSTRACT FROM AUTHOR]

Polar codes are the latest breakthrough in coding theory, as they are the first family of codes with explicit construction that provably achieve the symmetric capacity of binary-input discrete memoryless channels. Polar encoding and successive cancellation decoding have the complexities of $N \log N$ , for code length $N$ . Although, the complexity bound of $N \log N$ is asymptotically favorable, we report in this work methods to further reduce the encoding and decoding complexities of polar coding. The crux is to relax the polarization of certain bit-channels without performance degradation. We consider schemes for relaxing the polarization of both very good and very bad bit-channels, in the process of channel polarization. Relaxed polar codes are proved to preserve the capacity achieving property of polar codes. Analytical bounds on the asymptotic and finite-length complexity reduction attainable by relaxed polarization are derived. For binary erasure channels, we show that the computation complexity can be reduced by a factor of six, while preserving the rate and error performance. We also show that relaxed polar codes can be decoded with significantly reduced latency. For additive white Gaussian noise channels with medium code lengths, we show that relaxed polar codes can have lower error probabilities than conventional polar codes, while having reduced encoding and decoding computation complexities. [ABSTRACT FROM PUBLISHER]

Streaming codes take a string of source symbols as input and output a string of coded symbols in real time, which eliminate the queueing delay of traditional block codes and are thus especially appealing for delay sensitive applications. Existing works on streaming code performance either focused on the asymptotic error-exponent analyses, or on the optimal code construction under deterministic adversarial channel models. In contrast, this work analyzes the exact error probability of random linear streaming codes (RLSCs) in the large field size regime over the stochastic i.i.d. symbol erasure channel model. A closed-form expression of the error probability of large-field-size RLSCs is derived under, simultaneously, the finite memory length and decoding deadline constraints. The result is then used to examine the intricate tradeoff between memory length (complexity), decoding deadline (delay), code rate (throughput), and error probability (reliability). Numerical evaluation shows that under the same code rate and error probability requirements, the end-to-end delay of RLSCs is 40–48% of that of the optimal block codes (i.e., MDS codes). This implies that switching from block codes to streaming codes not only eliminates the queueing delay completely (which accounts for the initial 50% of the delay reduction) but also improves the reliability (which accounts for the additional 2–10% delay reduction). [ABSTRACT FROM AUTHOR]

Consider an interleave-division multi-user coding model for a Gaussian multiple-access channel (MAC). Each user’s message is encoded by a separate channel encoder followed by a user-interleaver, which is employed for user separation. All the users’ codewords are jointly regarded as one multi-user codeword, and the distance between two multi-user codewords is defined as the Euclidean distance after the inter-user superposition caused by the MAC. A distance enumerator analysis estimates the multi-user maximum likelihood decoding performance. By introducing user-scramblers, a multi-user code ensemble is defined, and its distance enumerator is calculated as concatenated multiple single-user codes and a superposition code. Both finite-length and asymptotic analyses are given and it is shown that interleave-division multi-user codes have much better distance properties than conventional random direct-sequence code-division multiple-access. In fact, the codes achieve almost the same asymptotic uniquely decodable performance as a multi-user random code. [ABSTRACT FROM PUBLISHER]

Opportunistic detection rules (ODRs) are variants of fixed-sample-size detection rules in which the statistician is allowed to make an early decision on the alternative hypothesis opportunistically based on the sequentially observed samples. From a sequential decision perspective, ODRs are also mixtures of one-sided and truncated sequential detection rules. Several results regarding ODRs are established in this paper. In the finite regime, the maximum sample size is modeled either as a fixed finite number, or a geometric random variable with a fixed finite mean. For both cases, the corresponding Bayesian formulations are investigated. The former case is a slight variation of the well-known finite-length sequential hypothesis testing procedure in the literature, whereas the latter case is new, for which the Bayesian optimal ODR is shown to be a sequence of likelihood ratio threshold tests with two different thresholds. A running threshold, which is determined by solving a stationary state equation, is used when future samples are still available, and a terminal threshold (simply the ratio between the priors scaled by costs) is used when the statistician reaches the final sample and, thus, has to make a decision immediately. In the asymptotic regime, the tradeoff among the exponents of the (false alarm and miss) error probabilities and the normalized expected stopping time under the alternative hypothesis is completely characterized and proved to be tight, via an information-theoretic argument. Within the tradeoff region, one noteworthy fact is that the performance of the Stein-Chernoff lemma is attainable by ODRs. [ABSTRACT FROM AUTHOR]

In this paper, we estimate the fidelity of stabilizer and CSS codes. First, we derive a lower bound on the fidelity of a stabilizer code via its quantum enumerator. Next, we find the average quantum enumerators of the ensembles of finite length stabilizer and CSS codes. We use the average quantum enumerators for obtaining lower bounds on the average fidelity of these ensembles. We further improve the fidelity bounds by estimating the quantum enumerators of expurgated ensembles of stabilizer and CSS codes. Finally, we derive fidelity bounds in the asymptotic regime when the code length tends to infinity. These results tell us which code rate we can afford for achieving a target fidelity with codes of a given length. The results also show that in symmetric depolarizing channel a typical stabilizer code has better performance, in terms of fidelity and code rate, compared with a typical CSS codes, and that balanced CSS codes significantly outperform other CSS codes. Asymptotic results demonstrate that CSS codes have a fundamental performance loss compared with stabilizer codes. [ABSTRACT FROM PUBLISHER]

Which communication rates can be attained over a channel whose output is an unknown (possibly stochastic) function of the input that may vary arbitrarily in time with no a priori model? Following the spirit of the finite-state compressibility of a sequence, defined by Lempel and Ziv, a “capacity” is defined for such a channel as the highest rate achievable by a designer knowing the particular relation that indeed exists between the input and output for all times, yet is constrained to use a fixed finite-length block communication scheme without feedback, i.e., use the same encoder and decoder over each block. In the case of the modulo additive channel, where the output sequence is obtained by modulo addition of an unknown individual sequence to the input sequence, this capacity is upper bounded by a function of the finite state compressibility of the noise sequence. A universal communication scheme with feedback that attains this capacity universally, without prior knowledge of the noise sequence, is presented. [ABSTRACT FROM AUTHOR]

A coding scheme for write once memory (WOM) using polar codes is presented. It is shown that the scheme achieves the capacity region of noiseless WOMs when an arbitrary number of multiple writes is permitted. The encoding and decoding complexities scale as O(N \log \; N), where N is the blocklength. For N sufficiently large, the error probability decreases subexponentially in N. The results can be generalized from binary to generalized WOMs, described by an arbitrary directed acyclic graph, using nonbinary polar codes. In the derivation, we also obtain results on the typical distortion of polar codes for lossy source coding. Some simulation results with finite length codes are presented. [ABSTRACT FROM AUTHOR]

Spatially coupled codes have been of interest recently owing to their superior performance over memoryless binary-input channels. The performance is good both asymptotically, since the belief propagation thresholds approach the Shannon limit, as well as for finite lengths, since degree-2 variable nodes that result in high error floors can be completely avoided. However, to realize the promised good performance, one needs large blocklengths. This in turn implies a large latency and decoding complexity. For the memoryless binary erasure channel, we consider the decoding of spatially coupled codes through a windowed decoder that aims to retain many of the attractive features of belief propagation, while trying to reduce complexity further. We characterize the performance of this scheme by defining thresholds on channel erasure rates that guarantee a target erasure rate. We give analytical lower bounds on these thresholds and show that the performance approaches that of belief propagation exponentially fast in the window size. We give numerical results including the thresholds computed using density evolution and the erasure rate curves for finite-length spatially coupled codes. [ABSTRACT FROM PUBLISHER]

We examine codes, over the additive Gaussian noise channel, designed for reliable communication at some specific signal-to-noise ratio (SNR) and constrained by the permitted minimum mean-square error (MMSE) at lower SNRs. The maximum possible rate is below point-to-point capacity, and hence, these are nonoptimal codes (alternatively referred to as “bad” codes). We show that the maximum possible rate is the one attained by superposition codebooks. Moreover, the MMSE and mutual information behavior as a function of SNR, for any code attaining the maximum rate under the MMSE constraint, is known for all SNR. We also provide a lower bound on the MMSE for finite length codes, as a function of the error probability of the code. [ABSTRACT FROM AUTHOR]