Conservation laws for coding

Spatially-coupled (SC) LDPC codes are constructed from a set of L regular sparse codes of length M. In the asymptotic limit of these parameters, SC codes present an excellent decoding threshold under belief propagation (BP) decoding, close to the maximum a posteriori (MAP) threshold of the underlying regular code. In the finite-length regime, we need both dimensions, L and M, to be sufficiently large, yielding a very large code length and decoding latency. In this paper, and for the erasure channel, we show that the finite-length performance of SC codes is improved if we consider the tree-structured expectation propagation (TEP) algorithm in the decoding stage. When applied to the decoding of SC LDPC codes, it allows using shorter codes to achieve similar error rates. We also propose a window-sliding scheme for the TEP decoder to reduce the decoding latency.

This paper addresses extrinsic information chart based design of non-systematic irregular repeat-accumulate codes for a given number of decoding iterations. This criterion is of practical importance in many applications where complexity or latency is limited. Our main contribution is a novel formulation of the optimzation problem in a particular way that makes its actual evaluation possible. This is achieved by introducing additional optimization variables and constraints which have the effect of “unrolling” the iteration. Our approach enforces the finite iteration condition, while avoiding the need to know arbitrary repeated functional compositions of the component EXIT functions. We restrict attention to the binary erasure channel, for which the EXIT chart approach is exact. Extension to other sparse-graph codes and other communication channels is straightforward.

In this letter, a new early termination method is proposed for Luby transform (LT) belief propagation (BP) decoder. The proposed method is decided whether BP decoder output converges to original data bits by observing only sign alterations of log-likelihood ratio (LLR) messages in BP decoder structure. Simulation results and complexity analyzes show that proposed method has low computational complexity and small average iteration amounts compared to conventional early termination methods in literature. In addition to this, the proposed method doesn't cause any performance degradation in BP decoder. The method can be easily applied to code families which can be decoded by BP algorithm such as low density parity check (LDPC) codes, polar codes and Raptor codes.

In this paper, we discuss the belief propagation (BP) decoding of polar codes. The performance of polar codes for short lengths is not satisfactory. Therefore, motivated by this we propose a novel technique to improve the performance of polar codes under BP decoding. In order to enhance the reliability of the variable nodes’ propagated messages in BP decoding, the messages are multiplied by previous messages of variable nodes. It is also shown that the BP decoding can be performed on polar codes if the factor graph is constructed according to the parity check matrix of polar codes. Simulation results in binary-input additive white Gaussian noise channel show that using the proposed method a gain of 1–2 dB can be achieved. We also show that the complexity of proposed decoder is same as the complexity of standard BP decoder. Furthermore, we show that the proposed decoder requires about 10–25 iterations less than the standard BP decoder.

Convolutional low-density parity-check (LDPC) ensembles, introduced by Felström and Zigangirov, have excellent thresholds and these thresholds are rapidly increasing functions of the average degree. Several variations on the basic theme have been proposed to date, all of which share the good performance characteristics of convolutional LDPC ensembles. We describe the fundamental mechanism that explains why “convolutional-like” or “spatially coupled” codes perform so well. In essence, the spatial coupling of individual codes increases the belief-propagation (BP) threshold of the new ensemble to its maximum possible value, namely the maximum a posteriori (MAP) threshold of the underlying ensemble. For this reason, we call this phenomenon “threshold saturation.” This gives an entirely new way of approaching capacity. One significant advantage of this construction is that one can create capacity-approaching ensembles with an error correcting radius that is increasing in the blocklength. Although we prove the “threshold saturation” only for a specific ensemble and for the binary erasure channel (BEC), empirically the phenomenon occurs for a wide class of ensembles and channels. More generally, we conjecture that for a large range of graphical systems a similar saturation of the “dynamical” threshold occurs once individual components are coupled sufficiently strongly. This might give rise to improved algorithms and new techniques for analysis.

We present the tree-structure expectation propagation (Tree-EP) algorithm to decode low-density parity-check (LDPC) codes over discrete memoryless channels (DMCs). EP 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 pair-wise 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 regime 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 (SL) for finite-length LDPC codes, which accurately approximates the TEP improved performance and facilitates its optimization.

We study the stability of low-density parity-check codes under blockwise or bitwise maximum a posteriori decoding, where transmission takes place over a binary-input memoryless output-symmetric channel. Our study stems from the consideration of constructing universal capacity-achieving codes under low-complexity decoding algorithms, where universality refers to the fact that we are considering a family of channels with equal capacity. Consider, e.g., the right-regular sequence by Shokrollahi and the heavy-tail Poisson sequence by Luby et al. Both sequences are provably capacity-achieving under belief propagation decoding when transmission takes place over the binary erasure channel. In this paper we show that many existing capacity-achieving sequences of low-density parity-check codes are not universal under belief propagation decoding. We reveal that the key to showing this non-universality result is determined by the stability of the underlying codes. More concretely, for an ordered and complete channel family and a sequence of low-density parity-check code ensembles, we determine a stability threshold associated with them, which gives rise to a sufficient condition under which the sequence is not universal under belief propagation decoding. Moreover, we show that the same stability threshold applies to blockwise or bitwise maximum a posteriori decoding as well. We demonstrate how the stability threshold can determine an upper bound on the corresponding blockwise or bitwise maximum a posteriori threshold, revealing the operational significance of the stability threshold.

This paper describes the analysis of convolutional codes on the erasure channel. We compare the maximum likelihood (ML) sequence decision and the maximum a posteriori (MAP) symbol decision for codes, which are transmitted over the erasure channel. When a codeword from a linear error correcting code with elements from the field GF is transmitted over a q-ary erasure channel, the symbol error rate of the maximum likelihood (ML) sequence decision is the same as that of the symbol maximum a posteriori (MAP) probability decision. When decoding convolutional codes transmitted over an AWGN channel, it is widely known that the probability of symbol error for the Viterbi algorithm (which is a sequence ML decoder) is generally higher than that for the more complex BCJR algorithm (which is a symbol MAP decoder).

A new iterative soft decision decoding method with very low complexity for Reed-Solomon (RS) codes is proposed. This method is based on bit level belief propagation (BP) decoding. In order to make BP decoding effective for RS codes, we use an extended binary parity check matrix with a lower density and reduced number of 4-cycles compared to the original binary parity check matrix of the code. Our proposed method is based on information correction in BP decoding. It means that we determine least reliable bits and by changing their channel information, the convergence of the decoder is improved. Our method results in considerable performance improvement of RS codes compared to hard decision decoding. The performance is also superior or comparable to some popular soft decision decoding methods.

Rateless coding such as Luby transform (LT) codes is representative of rate adaptive solutions to improve the frequency efficiency of transmitting data over a time-varying wireless channel. Also, these type of codes use belief propagation (BP) decoding which searches for encoded packets with degree 1 among the large numbers of encoded packets. This method fails if no encoded packets with degree 1 exist. However, in reality, small numbers of input packets are often encountered. In this paper, we have proposed a decoding algorithm that increases the performance of these codes for the small number of source packets. After the failure of BP decoding, to recover the remaining source packets, we use a method which keeps on decoding using the encoded packets whose degrees are not one. This algorithm continues until a degree 1 packet emerges in one of the iterations of this decoding process. Then, decoding algorithm switches back to the traditional BP decoding.

In this paper, decentralized message passing is addressed for a placement problem, named the minimum sensor cover problem: given point-wise tasks and sensors with omnidirectional coverage, find the minimum set of sensors that cover all the tasks. The minimum sensor cover problem can be converted to the geometric set cover by discretization of the solution space of the problem, and the geometric set cover is formulated as a maximum a posteriori (MAP) state assignment problem in a pairwise Markov random field model, which is a particular type of graphical models representing dependency relations between statistical variables. Belief propagation algorithm is a local iterative message passing algorithm devised to solve statistical inference problems in graphical models, but in a MAP assignment formulation of the set cover problem, solutions of belief propagation do not always result in convergence and the algorithm even produces infeasible solutions in largesized problems. A message passing algorithm based on belief propagation is presented to obtain stable and guaranteed feasible solutions by decentralized computations and numerical simulation validates the convergence, feasibility, and preferable solution quality of the presented algorithm against existing modifications of belief propagation and centralized greedy algorithm for the set cover.

In this paper, we put forward the tree-structured expectation propagation (TEP) algorithm for decoding block and convolutional low-density parity-check codes over any binary channel. We have already shown that TEP improves belief propagation (BP) over the binary erasure channel (BEC) by imposing marginal constraints over a set of pairs of variables that form a tree or a forest. The TEP decoder is a message-passing algorithm that sequentially builds a tree/forest of erased variables to capture additional information disregarded by the standard BP decoder, which leads to a noticeable reduction of the error rate for finite-length codes. In this paper, we show how the TEP can be extended to any channel, specifically to binary memoryless symmetric (BMS) channels. We particularly focus on how the TEP algorithm can be adapted for any channel model and, more importantly, how to choose the tree/forest to keep the gains observed for block and convolutional LDPC codes over the BEC.

This paper presents a general decoder architecture for 5G LDPC and polar codes. To achieve the hardware reuse maximization, both LDPC and polar codes are decoded by belief propagation (BP) algorithm on sparse bipartite graphs. However, by counting the number of short cycles on the polar bipartite graphs, we observe that numerous short cycles cause the poor performance than conventional decoding algorithms, like successive cancellation (SC) and BP decoding. To reduce the impact of short cycles, a LDPC-like BP list (BPL) decoding algorithm is proposed for polar codes, which is applied on L different polar bipartite graphs constructed by different permuted polar factor graphs. Finally, the simulation shows that the proposed general decoder for polar codes can obtain beyond SC and BP error performance with lower decoding latency and computational complexity, and even near successive cancellation list (SCL) performance in high signal-noise ratios (SNRs).

Consider transmission over the binary erasure channel using LDPC codes. We observed in [C. Measson, et al. (2003)] a curious relationship between the resulting maximum-likelihood (ML) decoding curve and the related performance curve under iterative (IT) decoding. We interpret this as an instance of Maxwell's construction which in its original form expresses an equilibrium condition at a phase transition.

Polar codes have been proved to be able to achieve the capacity of a Binary-Input Discrete Memoryless Channel (B-DMC) despite the possibility of a low-complexity encoding and decoding. Belief Propagation (BP) decoding is one of the major techniques proposed and vastly used for decoding polar codes, which has the advantage of high throughput compared with the other main decoding technique, namely Successive Cancellation (SC). The number of iterations in BP decoding is of utmost importance, as it directly impacts decoding performance, throughput, and power consumption. Devising an appropriate criterion for early termination of decoding once the remaining iterations are not necessary is therefore a useful technique for preventing further latency and power consumption. In this paper, an H-matrix-based criterion for early termination is introduced and shown to be more effective than the conventional G-matrix-based technique without sacrificing the Bit-Error Rate (BER) performance. It is also argued that the proposed method is simpler to be implemented, and its complexity is further reduced as the code-rate increases, unlike the G-matrix criterion whose complexity grows as the code-rate increases.