The generalized Poor-Verdu error lower bound established in [1] for multihypothesis testing is studied in the classical channel coding context. It is proved that for any sequence of block codes sent over the memoryless binary symmetric channel (BSC), the minimum probability of error (under maximum likelihood decoding) has a relative deviation from the generalized bound that grows at most linearly in blocklength. This result directly implies that for arbitrary codes used over the BSC, decoder ties can only affect the subexponential behavior of the minimum probability of error.
Consider a discrete memoryless multiple source with $m$ components of which $k \leq m$ possibly different sources are sampled at each time instant and jointly compressed in order to reconstruct all the $m$ sources under a given distortion criterion. A new notion of sampling rate distortion function is introduced, and is characterized first for the case of fixed-set sampling. Next, for independent random sampling performed without knowledge of the source outputs, it is shown that the sampling rate distortion function is the same regardless of whether or not the decoder is informed of the sequen…
Polar codes under successive cancellation decoding proposed by Arıkan provably achieve the symmetric capacity of any given binary-input discrete memoryless channel. The successive cancellation list decoder for polar codes was described by Tal and Vardy as a generalization of the successive cancellation decoder of Arıkan. The performance of the successive cancellation list decoder is encouraging in practice. In this paper, we formalize the successive cancellation list decoder in our notation and prove that polar codes under successive cancellation list decoding achieve the symmetric capacity of…
Motivated by applications of biometric identification and content identification systems, we consider the problem of random coding for channels, where each codeword undergoes lossy compression (vector quantization), and where the decoder bases its decision only on the compressed codewords and the channel output, which is in turn, the channel's response to the transmission of an original codeword, before compression. For memoryless sources and memoryless channels with finite alphabets, we propose a new universal decoder and analyze its error exponent, which improves on an earlier result by Dasa…
Physical-layer Network Coding (PNC) can significantly improve the throughput of two-way relay channels. An interesting variant of PNC is Analog Network Coding (ANC). Almost all ANC schemes proposed to date, however, operate in a symbol by symbol manner (memoryless) and cannot exploit the redundant information in channel-coded packets to enhance performance. This paper proposes a non-memoryless ANC scheme. In particular, we design a soft-input soft-output decoder for the relay node to process the superimposed packets from the two end nodes to yield an estimated MMSE packet for forwarding back t…
In this paper, we propose a novel joint coding-modulation technique based on serial concatenation of orthogonal linear transform, such as discrete Fourier transform (DFT) or Walsh-Hadamard transform (WHT), with memoryless nonlinearity. We demonstrate that such a simple signal construction may exhibit properties of a random code ensemble, as a result approaching channel capacity. Our computer simulations confirm that if the decoder relies on a modified approximate message passing algorithm, the proposed modulation technique exhibits performance on par with state-of-the-art coded modulation sche…
In this paper, we give a distributed joint source channel coding scheme for arbitrary correlated sources for arbitrary point in the Slepian-Wolf rate region, and arbitrary link capacities using LDPC codes. We consider the Slepian-Wolf setting of two sources and one destination, with one of the sources derived from the other source by some correlation model known at the decoder. Distributed encoding and separate decoding is used for the two sources. We also give a distributed source coding scheme when the source correlation has memory to achieve any point in the Slepian-Wolf rate achievable reg…
We generalize the problem of controlling the interference created to an external observer while communicating over a discrete memoryless channel (DMC) which was studied in \cite{serrano:2014}. In particular, we consider the scenario where the transmission is established over a compound DMC channel with unknown state at both the encoder and the decoder. Depending on the exact state $s$ of the channel, we ask for a different level of average precision $Δ_s$ on the establishment of the interference coordination with the external observer. For this set-up, we fully characterize the capacity region…
Arıkan's polar coding, is by now a well studied technique that allows achieving the symmetric capacity of binary input memoryless channels with low complexity encoding and decoding, provided that the polar decoding architecture is used and the decoding metric is matched to the true channel. In this paper, we analyze communication rates that are achievable when the polar coding/decoding architecture is used with the decoder using an incorrect model of the channel. We define the `polar mismatched capacity' as an analogue of the classical mismatched capacity, give an expression for it, and derive…
Polar codes provably achieve the symmetric capacity of a memoryless channel while having an explicit construction. This work aims to increase the throughput of polar decoder hardware by an order of magnitude relative to the state of the art successive-cancellation decoder. We present an algorithm, architecture, and FPGA implementation of a gigabit-per-second polar decoder.
We study the performance limits of state-dependent discrete memoryless channels with a discrete state available at both the encoder and the decoder. We establish the epsilon-capacity as well as necessary and sufficient conditions for the strong converse property for such channels when the sequence of channel states is not necessarily stationary, memoryless or ergodic. We then seek a finer characterization of these capacities in terms of second-order coding rates. The general results are supplemented by several examples including i.i.d. and Markov states and mixed channels.
We propose a source/channel duality in the exponential regime, where success/failure in source coding parallels error/correctness in channel coding, and a distortion constraint becomes a log-likelihood ratio (LLR) threshold. We establish this duality by first deriving exact exponents for lossy coding of a memoryless source P, at distortion D, for a general i.i.d. codebook distribution Q, for both encoding success (R < R(P,Q,D)) and failure (R > R(P,Q,D)). We then turn to maximum likelihood (ML) decoding over a memoryless channel P with an i.i.d. input Q, and show that if we substitute P=QP, Q=…
We consider the problem of communicating over the general discrete memoryless broadcast channel (BC) with partially cooperating receivers. In our setup, receivers are able to exchange messages over noiseless conference links of finite capacities, prior to decoding the messages sent from the transmitter. In this paper we formulate the general problem of broadcast with cooperation. We first find the capacity region for the case where the BC is physically degraded. Then, we give achievability results for the general broadcast channel, for both the two independent messages case and the single comm…
A new single-letter achievable rate region is proposed for the two-user discrete memoryless multiple-access channel(MAC) with noiseless feedback. The proposed region includes the Cover-Leung rate region [1], and it is shown that the inclusion is strict. The proof uses a block-Markov superposition strategy based on the observation that the messages of the two users are correlated given the feedback. The rates of transmission are too high for each encoder to decode the other's message directly using the feedback, so they transmit correlated information in the next block to learn the message of o…
We show that the Extrinsic Information about the coded bits of any good (capacity achieving) code operating over a wide class of discrete memoryless channels (DMC) is zero when channel capacity is below the code rate and positive constant otherwise, that is, the Extrinsic Information Transfer (EXIT) chart is a step function of channel quality, for any capacity achieving code. It follows that, for a common class of iterative receivers where the error correcting decoder must operate at first iteration at rate above capacity (such as in turbo equalization, turbo channel estimation, parallel and s…
We consider the problem of erasure/list decoding using certain classes of simplified decoders. Specifically, we assume a class of erasure/list decoders, such that a codeword is in the list if its likelihood is larger than a threshold. This class of decoders both approximates the optimal decoder of Forney, and also includes the following simplified subclasses of decoding rules: The first is a function of the output vector only, but not the codebook (which is most suitable for high rates), and the second is a scaled version of the maximum likelihood decoder (which is most suitable for low rates)…
We consider a decoder with an erasure option and a variable size list decoder for channels with non-casual side information at the transmitter. First, universally achievable error exponents are offered for decoding with an erasure option using a parameterized decoder in the spirit of Csiszár and Körner's decoder. Then, the proposed decoding rule is generalized by extending the range of its parameters to allow variable size list decoding. This extension gives a unified treatment for erasure/list decoding. Exponential bounds on the probability of list error and the average number of incorrect me…
The optimal decoder achieving the outage capacity under imperfect channel estimation is investigated. First, by searching into the family of nearest neighbor decoders, which can be easily implemented on most practical coded modulation systems, we derive a decoding metric that minimizes the average of the transmission error probability over all channel estimation errors. This metric, for arbitrary memoryless channels, achieves the capacity of a composite (more noisy) channel. Next, according to the notion of estimation-induced outage capacity (EIO capacity) introduced in our previous work, we c…
We show that the mismatched capacity of binary discrete memoryless channels can be improved by channel combining and splitting via Arıkan's polar transformations. We also show that the improvement is possible even if the transformed channels are decoded with a mismatched polar decoder.
The security level of the achievability scheme for Wyner's wiretap channel model is examined from the perspective of the probability of correct decoding, $P_c$, at the wiretap channel decoder. In particular, for finite-alphabet memoryless channels, the exact random coding exponent of $P_c$ is derived as a function of the total coding rate $R_1$ and the rate of each sub-code $R_2$. Two different representations are given for this function and its basic properties are provided. We also characterize the region of pairs of rates $(R_1,R_2)$ of full security in the sense of the random coding expone…
Sparse superposition (SS) codes were originally proposed as a capacity-achieving communication scheme over the additive white Gaussian noise channel (AWGNC) [1]. Very recently, it was discovered that these codes are universal, in the sense that they achieve capacity over any memoryless channel under generalized approximate message-passing (GAMP) decoding [2], although this decoder has never been stated for SS codes. In this contribution we introduce the GAMP decoder for SS codes, we confirm empirically the universality of this communication scheme through its study on various channels and we p…
Polar codes are able to achieve the capacity of memoryless channels under successive cancellation (SC) decoding. Soft Cancellation (SCAN) is a soft-output decoder based on the SC schedule, useful in iterative decoding and concatenation of polar codes. However, the sequential nature of this decoder leads to high decoding latency compared to state-of-the-art codes. To reduce the latency of SCAN, in this paper we identify special nodes in the decoding tree, corresponding to specific frozen-bit sequences, and propose dedicated low-latency decoding approaches for each of them. The resulting fast-SC…
We present an analysis, under iterative decoding, of coset LDPC codes over GF(q), designed for use over arbitrary discrete-memoryless channels (particularly nonbinary and asymmetric channels). We use a random-coset analysis to produce an effect that is similar to output-symmetry with binary channels. We show that the random selection of the nonzero elements of the GF(q) parity-check matrix induces a permutation-invariance property on the densities of the decoder messages, which simplifies their analysis and approximation. We generalize several properties, including symmetry and stability from …
The recently-discovered polar codes are seen as a major breakthrough in coding theory; they provably achieve the theoretical capacity of discrete memoryless channels using the low complexity successive cancellation (SC) decoding algorithm. Motivated by recent developments in polar coding theory, we propose a family of efficient hardware implementations for SC polar decoders. We show that such decoders can be implemented with O(n) processing elements, O(n) memory elements, and can provide a constant throughput for a given target clock frequency. Furthermore, we show that SC decoding can be impl…
A real time coding system with lookahead consists of a memoryless source, a memoryless channel, an encoder, which encodes the source symbols sequentially with knowledge of future source symbols upto a fixed finite lookahead, d, with or without feedback of the past channel output symbols and a decoder, which sequentially constructs the source symbols using the channel output. The objective is to minimize the expected per-symbol distortion. For a fixed finite lookahead d>=1 we invoke the theory of controlled markov chains to obtain an average cost optimality equation (ACOE), the solution of whic…
Communication systems are usually designed by assuming perfect channel state information (CSI). However, in many practical scenarios, only a noisy estimate of the channel is available, which may strongly differ from the true channel. This imperfect CSI scenario is addressed by introducing the notion of estimation-induced outage (EIO) capacity. We derive a single-letter characterization of the maximal EIO rate and prove an associated coding theorem and its strong converse for discrete memoryless channels (DMCs). The transmitter and the receiver rely on the channel estimate and the statistics of…
There is a fundamental relationship between belief propagation and maximum a posteriori decoding. The case of transmission over the binary erasure channel was investigated in detail in a companion paper. This paper investigates the extension to general memoryless channels (paying special attention to the binary case). An area theorem for transmission over general memoryless channels is introduced and some of its many consequences are discussed. We show that this area theorem gives rise to an upper-bound on the maximum a posteriori threshold for sparse graph codes. In situations where this boun…
Posterior matching (PM) is a sequential horizon-free feedback communication scheme introduced by the authors, who also provided a rather involved optimality proof showing it achieves capacity for a large class of memoryless channels. Naghshvar et al considered a non-sequential variation of PM with a fixed number of messages and a random decision-time, and gave a simpler proof establishing its optimality via a novel Shannon-Jensen divergence argument. Another simpler optimality proof was given by Li and El Gamal, who considered a fixed-rate fixed block-length variation of PM with an additional …
A method is proposed, called channel polarization, to construct code sequences that achieve the symmetric capacity $I(W)$ of any given binary-input discrete memoryless channel (B-DMC) $W$. The symmetric capacity is the highest rate achievable subject to using the input letters of the channel with equal probability. Channel polarization refers to the fact that it is possible to synthesize, out of $N$ independent copies of a given B-DMC $W$, a second set of $N$ binary-input channels $\{W_N^{(i)}:1\le i\le N\}$ such that, as $N$ becomes large, the fraction of indices $i$ for which $I(W_N^{(i)})$ …
A coded modulation system is considered in which nonbinary coded symbols are mapped directly to nonbinary modulation signals. It is proved that if the modulator-channel combination satisfies a particular symmetry condition, the codeword error rate performance is independent of the transmitted codeword. It is shown that this result holds for both linear-programming decoders and sum-product decoders. In particular, this provides a natural modulation mapping for nonbinary codes mapped to PSK constellations for transmission over memoryless channels such as AWGN channels or flat fading channels wit…
Polar codes are of great interests because they provably achieve the capacity of both discrete and continuous memoryless channels while having an explicit construction. Most existing decoding algorithms of polar codes are based on bit-wise hard or soft decisions. In this paper, we propose symbol-decision successive cancellation (SC) and successive cancellation list (SCL) decoders for polar codes, which use symbol-wise hard or soft decisions for higher throughput or better error performance. First, we propose to use a recursive channel combination to calculate symbol-wise channel transition pro…
We propose a new class of efficient decoding algorithms for Reed-Muller (RM) codes over binary-input memoryless channels. The algorithms are based on projecting the code on its cosets, recursively decoding the projected codes (which are lower-order RM codes), and aggregating the reconstructions (e.g., using majority votes). We further provide extensions of the algorithms using list-decoding. We run our algorithm for AWGN channels and Binary Symmetric Channels at the short code length ($\le 1024$) regime for a wide range of code rates. Simulation results show that in both low code rate and high…
This paper investigates the joint source-channel coding problem of sending a memoryless source over a memoryless broadcast channel. An inner bound and several outer bounds on the admissible distortion region are derived, which respectively generalize and unify several existing bounds. As a consequence, we also obtain an inner bound and an outer bound for the degraded broadcast channel case. When specialized to the Gaussian or binary source broadcast, the inner bound and outer bound not only recover the best known inner bound and outer bound in the literature, but also generate some new results…
The problem of state communication over a discrete memoryless channel with discrete memoryless state is studied when the state information is available strictly causally at the encoder. It is shown that block Markov encoding, in which the encoder communicates a description of the state sequence in the previous block by incorporating side information about the state sequence at the decoder, yields the minimum state estimation error. When the same channel is used to send additional independent information at the expense of a higher channel state estimation error, the optimal tradeoff between the…
Long polar codes can achieve the symmetric capacity of arbitrary binary-input discrete memoryless channels under a low complexity successive cancelation (SC) decoding algorithm. However, for polar codes with short and moderate code length, the decoding performance of the SC algorithm is inferior. The cyclic redundancy check (CRC) aided successive cancelation list (SCL) decoding algorithm has better error performance than the SC algorithm for short or moderate polar codes. In this paper, we propose an efficient list decoder architecture for the CRC aided SCL algorithm, based on both algorithmic…
There is a fundamental relationship between belief propagation and maximum a posteriori decoding. A decoding algorithm, which we call the Maxwell decoder, is introduced and provides a constructive description of this relationship. Both, the algorithm itself and the analysis of the new decoder are reminiscent of the Maxwell construction in thermodynamics. This paper investigates in detail the case of transmission over the binary erasure channel, while the extension to general binary memoryless channels is discussed in a companion paper.
Polar codes, invented by Arikan in 2009, are known to achieve the capacity of any binary-input memoryless output-symmetric channel. One of the few drawbacks of the original polar code construction is that it is not universal. This means that the code has to be tailored to the channel if we want to transmit close to capacity. We present two "polar-like" schemes which are capable of achieving the compound capacity of the whole class of binary-input memoryless output-symmetric channels with low complexity. Roughly speaking, for the first scheme we stack up $N$ polar blocks of length $N$ on top of…
The subject of this paper is transmission over a general class of binary-input memoryless symmetric channels using error correcting codes based on sparse graphs, namely low-density generator-matrix and low-density parity-check codes. The optimal (or ideal) decoder based on the posterior measure over the code bits, and its relationship to the sub-optimal belief propagation decoder, are investigated. We consider the correlation (or covariance) between two codebits, averaged over the noise realizations, as a function of the graph distance, for the optimal decoder. Our main result is that this cor…
Over discrete memoryless channels (DMC), linear decoders (maximizing additive metrics) afford several nice properties. In particular, if suitable encoders are employed, the use of decoding algorithm with manageable complexities is permitted. Maximum likelihood is an example of linear decoder. For a compound DMC, decoders that perform well without the channel's knowledge are required in order to achieve capacity. Several such decoders have been studied in the literature. However, there is no such known decoder which is linear. Hence, the problem of finding linear decoders achieving capacity for…
We consider a windowed decoding scheme for LDPC convolutional codes that is based on the belief-propagation (BP) algorithm. We discuss the advantages of this decoding scheme and identify certain characteristics of LDPC convolutional code ensembles that exhibit good performance with the windowed decoder. We will consider the performance of these ensembles and codes over erasure channels with and without memory. We show that the structure of LDPC convolutional code ensembles is suitable to obtain performance close to the theoretical limits over the memoryless erasure channel, both for the BP dec…
We consider a channel-independent decoder which is for i.i.d. random codes what the maximum mutual-information decoder is for constant composition codes. We show that this decoder results in exactly the same i.i.d. random coding error exponent and almost the same correct-decoding exponent for a given codebook distribution as the maximum-likelihood decoder. We propose an algorithm for computation of the optimal correct-decoding exponent which operates on the corresponding expression for the channel-independent decoder. The proposed algorithm comes in two versions: computation at a fixed rate an…
We examine regular and irregular repeat-accumulate (RA) codes with repetition degrees which are all even. For these codes and with a particular choice of an interleaver, we give an upper bound on the decoding error probability of a linear-programming based decoder which is an inverse polynomial in the block length. Our bound is valid for any memoryless, binary-input, output-symmetric (MBIOS) channel. This result generalizes the bound derived by Feldman et al., which was for regular RA(2) codes.
The discrete memoryless interference channel is modelled as a conditional probability distribution with two outputs depending on two inputs and has widespread applications in practical communication scenarios. In this paper, we introduce and study the quantum interference channel, a generalization of a two-input, two-output memoryless channel to the setting of quantum Shannon theory. We discuss three different coding strategies and obtain corresponding achievable rate regions for quantum interference channels. We calculate the capacity regions in the special cases of "very strong" and "strong"…
We study the following semi-deterministic setting of the joint source-channel coding problem: a deterministic source sequence (a.k.a. individual sequence) is transmitted via a memoryless channel, using delay-limited encoder and decoder, which are both implementable by periodically-varying finite-state machines, and the decoder is granted with access to side information, which is a noisy version of the source sequence. We first derive a lower bound on the achievable expected distortion in terms of the empirical statistics of the source sequence, the number of states of the encoder, the number o…
When information is to be transmitted over an unknown, possibly unreliable channel, an erasure option at the decoder is desirable. Using constant-composition random codes, we propose a generalization of Csiszar and Korner's Maximum Mutual Information decoder with erasure option for discrete memoryless channels. The new decoder is parameterized by a weighting function that is designed to optimize the fundamental tradeoff between undetected-error and erasure exponents for a compound class of channels. The class of weighting functions may be further enlarged to optimize a similar tradeoff for lis…
This paper focuses on finite-dimensional upper and lower bounds on decodable thresholds of Zm and binary low-density parity-check (LDPC) codes, assuming belief propagation decoding on memoryless channels. A concrete framework is presented, admitting systematic searches for new bounds. Two noise measures are considered: the Bhattacharyya noise parameter and the soft bit value for a maximum a posteriori probability (MAP) decoder on the uncoded channel. For Zm LDPC codes, an iterative m-dimensional bound is derived for m-ary-input/symmetric-output channels, which gives a sufficient stability cond…
We study the problem of universal decoding for unknown discrete memoryless channels in the presence of erasure/list option at the decoder, in the random coding regime. Specifically, we harness a universal version of Forney's classical erasure/list decoder developed in earlier studies, which is based on the competitive minimax methodology, and guarantees universal achievability of a certain fraction of the optimum random coding error exponents. In this paper, we derive an exact single-letter expression for the maximum achievable fraction. Examples are given in which the maximal achievable fract…
We investigate the decoding region for Algebraic Soft-Decision Decoding (ASD) of Reed-Solomon codes in a discrete, memoryless, additive-noise channel. An expression is derived for the error correction radius within which the soft-decision decoder produces a list that contains the transmitted codeword. The error radius for ASD is shown to be larger than that of Guruswami-Sudan hard-decision decoding for a subset of low-rate codes. These results are also extended to multivariable interpolation in the sense of Parvaresh and Vardy. An upper bound is then presented for ASD's probability of error, w…
In this study we consider rateless coding over discrete memoryless channels (DMC) with feedback. Unlike traditional fixed-rate codes, in rateless codes each codeword is infinitely long, and the decoding time depends on the confidence level of the decoder. Using rateless codes along with sequential decoding, and allowing a fixed probability of error at the decoder, we obtain results for several communication scenarios. The results shown here are non-asymptotic, in the sense that the size of the message set is finite. First we consider the transmission of equiprobable messages using rateless cod…
With advancements in telecommunications, data transmission over increasingly harsher channels that produce synchronisation errors is inevitable. Coding schemes for such channels are available through techniques such as the Davey-MacKay watermark coding; however, this is limited to memoryless channel estimates. Memory must be accounted for to ensure a realistic channel approximation - similar to a Finite State Markov Chain or Fritchman Model. A novel code construction and decoder are developed to correct synchronisation errors while considering the channel's correlated memory effects by incorpo…
We consider a setting of Slepian--Wolf coding, where the random bin of the source vector undergoes channel coding, and then decoded at the receiver, based on additional side information, correlated to the source. For a given distribution of the randomly selected channel codewords, we propose a universal decoder that depends on the statistics of neither the correlated sources nor the channel, assuming first that they are both memoryless. Exact analysis of the random-binning/random-coding error exponent of this universal decoder shows that it is the same as the one achieved by the optimal maximu…
