Ensemble models are widely used to solve complex tasks by their decomposition into multiple simpler tasks, each one solved locally by a single member of the ensemble. Decoding of error-correction codes is a hard problem due to the curse of dimensionality, leading one to consider ensembles-of-decoders as a possible solution. Nonetheless, one must take complexity into account, especially in decoding. We suggest a low-complexity scheme where a single member participates in the decoding of each word. First, the distribution of feasible words is partitioned into non-overlapping regions. Thereafter, specialized experts are formed by independently training each member on a single region. A classical hard-decision decoder (HDD) is employed to map every word to a single expert in an injective manner. FER gains of up to 0.4dB at the waterfall region, and of 1.25dB at the error floor region are achieved for two BCH(63,36) and (63,45) codes with cycle-reduced parity-check matrices, compared to the previous best result of the paper "Active Deep Decoding of Linear Codes".
The performance of maximum-likelihood (ML) decoding on the binary erasure channel for finite-length low-density parity-check (LDPC) codes from two random ensembles is studied. The theoretical average spectrum of the Gallager ensemble is computed by using a recurrent procedure and compared to the empirically found average spectrum for the same ensemble as well as to the empirical average spectrum of the Richardson-Urbanke ensemble and spectra of selected codes from both ensembles. Distance properties of the random codes from the Gallager ensemble are discussed. A tightened union-type upper boun…
A ternary message passing (TMP) decoding algorithm for low-density parity-check codes is developed. All messages exchanged between variable and check nodes have a ternary alphabet, and the variable nodes exploit soft information from the channel. A density evolution analysis is developed for unstructured and protograph-based ensembles. For unstructured ensembles the stability condition is derived. Optimized ensembles for TMP decoding show asymptotic gains of up to 0.6 dB with respect to ensembles optimized for binary message passing decoding. Finite length simulations of codes from TMP-optimiz…
We consider automorphism ensemble decoding (AED) of quasi-cyclic (QC) low-density parity-check (LDPC) codes. Belief propagation (BP) decoding on the conventional factor graph is equivariant to the quasi-cyclic automorphisms and therefore prevents gains by AED. However, by applying small modifications to the parity-check matrix at the receiver side, we can break the symmetry without changing the code at the transmitter. This way, we can leverage a gain in error-correcting performance using an ensemble of identical BP decoders, without increasing the worst-case decoding latency. The proposed met…
In this paper we deal with polar code automorphisms that are beneficial under low-latency automorphism ensemble (AE) decoding, and we propose polar code designs that have such automorphisms. Successive-cancellation (SC) decoding and thus SC-based AE decoding are invariant with respect to the only known polar code automorphisms, namely those of the lower-triangular affine (LTA) group. To overcome this problem, we provide methods to determine whether a given polar code has non-LTA automorphisms and to identify such automorphisms. Building on this, we design specific polar codes that admit automo…
In this paper, we propose an analysis of the automorphism group of polar codes, with the scope of designing codes tailored for automorphism ensemble (AE) decoding. We prove the equivalence between the notion of decreasing monomial codes and the universal partial order (UPO) framework for the description of polar codes. Then, we analyze the algebraic properties of the affine automorphisms group of polar codes, providing a novel description of its structure and proposing a classification of automorphisms providing the same results under permutation decoding. Finally, we propose a method to list …
We analyze a class of high performance, low decoding-data-flow error-correcting codes suitable for high bit-rate optical-fiber communication systems. A spatially-coupled split-component ensemble is defined, generalizing from the most important codes of this class, staircase codes and braided block codes, and preserving a deterministic partitioning of component-code bits over code blocks. Our analysis focuses on low-complexity iterative algebraic decoding, which, for the binary erasure channel, is equivalent to a generalization of the peeling decoder. Using the differential equation method, we …
Polar codes have promising error-correction capabilities. Yet, decoding polar codes is often challenging, particularly with large blocks, with recently proposed decoders based on list-decoding or neural-decoding. The former applies multiple decoders or the same decoder multiple times with some redundancy, while the latter family utilizes emerging deep learning schemes to learn to decode from data. In this work we introduce a novel polar decoder that combines the list-decoding with neural-decoding, by forming an ensemble of multiple weighted belief-propagation (WBP) decoders, each trained to de…
A new approach for designing bilayer and multi-layer LDPC codes is proposed and studied in the asymptotic regime. The ensembles are defined through individual uni-variate degree distributions, one for each layer. We present a construction that: 1) enables low-complexity decoding for high-SNR channel instances, 2) provably approaches capacity for low-SNR instances, 3) scales linearly (in terms of design complexity) in the number of layers. For the setup where decoding the second layer is significantly more costly than the first layer, we propose an optimal-cost decoding schedule and study the t…
Reed-Muller (RM) codes are known for their good maximum likelihood (ML) performance in the short block-length regime. Despite being one of the oldest classes of channel codes, finding a low complexity soft-input decoding scheme is still an open problem. In this work, we present a versatile decoding architecture for RM codes based on their rich automorphism group. The decoding algorithm can be seen as a generalization of multiple-bases belief propagation (MBBP) and may use any polar or RM decoder as constituent decoders. We provide extensive error-rate performance simulations for successive can…
We consider ensembles of channel codes that are partitioned into bins, and focus on analysis of exact random coding error exponents associated with optimum decoding of the index of the bin to which the transmitted codeword belongs. Two main conclusions arise from this analysis: (i) for independent random selection of codewords within a given type class, the random coding exponent of optimal bin index decoding is given by the ordinary random coding exponent function, computed at the rate of the entire code, independently of the exponential rate of the size of the bin. (ii) for this ensemble of …
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…
In this paper we consider the generalization of binary spatially coupled low-density parity-check (SC-LDPC) codes to finite fields GF$(q)$, $q\geq 2$, and develop design rules for $q$-ary SC-LDPC code ensembles based on their iterative belief propagation (BP) decoding thresholds, with particular emphasis on low-latency windowed decoding (WD). We consider transmission over both the binary erasure channel (BEC) and the binary-input additive white Gaussian noise channel (BIAWGNC) and present results for a variety of $(J,K)$-regular SC-LDPC code ensembles constructed over GF$(q)$ using protographs…
In this paper we study the iterative decoding threshold performance of non-binary spatially-coupled low-density parity-check (NB-SC-LDPC) code ensembles for both the binary erasure channel (BEC) and the binary-input additive white Gaussian noise channel (BIAWGNC), with particular emphasis on windowed decoding (WD). We consider both (2,4)-regular and (3,6)-regular NB-SC-LDPC code ensembles constructed using protographs and compute their thresholds using protograph versions of NB density evolution and NB extrinsic information transfer analysis. For these code ensembles, we show that WD of NB-SC-…
Affine Subcode Ensemble Decoding for Degeneracy-Aware Quantum Error Correction
Communications Engineering Lab (CEL), Karlsruhe Institute of Technology (KIT)（卡尔斯鲁厄理工学院通信工程实验室）；School of Informatics, University of Edinburgh（爱丁堡大学信息学院）
Using the EXIT chart approach, a necessary and sufficient condition is developed for the local stability of iterative decoding of multi-edge type (MET) doubly-generalized low-density parity-check (D-GLDPC) code ensembles. In such code ensembles, the use of arbitrary linear block codes as component codes is combined with the further design of local Tanner graph connectivity through the use of multiple edge types. The stability condition for these code ensembles is shown to be succinctly described in terms of the value of the spectral radius of an appropriately defined polynomial matrix.
This paper is focused on the derivation of some universal properties of capacity-approaching low-density parity-check (LDPC) code ensembles whose transmission takes place over memoryless binary-input output-symmetric (MBIOS) channels. Properties of the degree distributions, graphical complexity and the number of fundamental cycles in the bipartite graphs are considered via the derivation of information-theoretic bounds. These bounds are expressed in terms of the target block/ bit error probability and the gap (in rate) to capacity. Most of the bounds are general for any decoding algorithm, and…
We analyze the finite-length performance of spatially coupled low-density parity-check (SC-LDPC) codes under window decoding over the binary erasure channel. In particular, we propose a refinement of the scaling law by Olmos and Urbanke for the frame error rate (FER) of terminated SC-LDPC ensembles under full belief propagation (BP) decoding. The refined scaling law models the decoding process as two independent Ornstein-Uhlenbeck processes, in correspondence to the two decoding waves that propagate toward the center of the coupled chain for terminated SC-LDPC codes. We then extend the propose…
A new method for low-complexity near-maximum-likelihood (ML) decoding of low-density parity-check (LDPC) codes over the additive white Gaussian noise channel is presented. The proposed method termed belief-propagation--list erasure decoding (BP-LED) is based on erasing carefully chosen unreliable bits performed in case of BP decoding failure. A strategy of introducing erasures into the received vector and a new erasure decoding algorithm are proposed. The new erasure decoding algorithm, called list erasure decoding, combines ML decoding over the BEC with list decoding applied if the ML decoder…
In this paper, we consider a few iterative decoding schemes for the joint source-channel coding of correlated sources. Specifically, we consider the joint source-channel coding of two erasure correlated sources with transmission over different erasure channels. Our main interest is in determining whether or not various code ensembles can achieve the capacity region universally over varying channel conditions. We consider two ensembles in the class of low-density generator-matrix (LDGM) codes known as Luby-Transform (LT) codes and one ensemble of low-density parity-check (LDPC) codes. We analyz…
Families of generalized spatially-coupled low-density parity-check (GSC-LDPC) code ensembles can be formed by terminating protograph-based generalized LDPC convolutional (GLDPCC) codes. It has previously been shown that ensembles of GSC-LDPC codes constructed from a protograph have better iterative decoding thresholds than their block code counterparts, and that, for large termination lengths, their thresholds coincide with the maximum a-posteriori (MAP) decoding threshold of the underlying generalized LDPC block code ensemble. Here we show that, in addition to their excellent iterative decodi…
We consider probabilistic shaping to maximize the achievable information rate of coded modulation (CM) with hard decision decoding. The proposed scheme using binary staircase codes outperforms its uniform CM counterpart by more than 1.3 dB for 64-QAM and 5 bits/symbol.
For a given family of spatially coupled codes, we prove that the LP threshold on the BSC of the graph cover ensemble is the same as the LP threshold on the BSC of the derived spatially coupled ensemble. This result is in contrast with the fact that the BP threshold of the derived spatially coupled ensemble is believed to be larger than the BP threshold of the graph cover ensemble as noted by the work of Kudekar et al. (2011, 2012). To prove this, we establish some properties related to the dual witness for LP decoding which was introduced by Feldman et al. (2007) and simplified by Daskalakis e…
A novel code construction based on spatially coupled low-density parity-check (SC-LDPC) codes is presented. The proposed code ensembles are described by protographs, comprised of several protograph-based chains characterizing individual SC-LDPC codes. We demonstrate that code ensembles obtained by connecting appropriately chosen SC-LDPC code chains at specific points have improved iterative decoding thresholds compared to those of single SC-LDPC coupled chains. In addition, it is shown that the improved decoding properties of the connected ensembles result in reduced decoding complexity requir…
We show how the iterative decoding threshold of tailbiting spatially coupled (SC) low-density parity-check (LDPC) code ensembles can be improved over the binary input additive white Gaussian noise channel by allowing the use of different transmission energies for the codeword bits. We refer to the proposed approach as energy shaping. We focus on the special case where the transmission energy of a bit is selected among two values, and where a contiguous portion of the codeword is transmitted with the largest one. Given these constraints, an optimal energy boosting policy is derived by means of …
A lower bound on the maximum likelihood (ML) decoding error exponent of linear block code ensembles, on the erasure channel, is developed. The lower bound turns to be positive, over an ensemble specific interval of erasure probabilities, when the ensemble weight spectral shape function tends to a negative value as the fractional codeword weight tends to zero. For these ensembles we can therefore lower bound the block-wise ML decoding threshold. Two examples are presented, namely, linear random parity-check codes and fixed-rate Raptor codes with linear random precoders. While for the former a f…
The binary primitive BCH codes are cyclic and are constructed by choosing a subset of the cyclotomic cosets. Which subset is chosen determines the dimension, the minimum distance and the weight distribution of the BCH code. We construct possible BCH codes and determine their coderate, true minimum distance and the non-equivalent codes. A particular choice of cyclotomic cosets gives BCH codes which are, extended by one bit, equivalent to Reed-Muller codes, which is a known result from the sixties. We show that BCH codes have possibly better parameters than Reed-Muller codes, which are related i…
We investigate spatially coupled code ensembles. For transmission over the binary erasure channel, it was recently shown that spatial coupling increases the belief propagation threshold of the ensemble to essentially the maximum a-priori threshold of the underlying component ensemble. This explains why convolutional LDPC ensembles, originally introduced by Felstrom and Zigangirov, perform so well over this channel. We show that the equivalent result holds true for transmission over general binary-input memoryless output-symmetric channels. More precisely, given a desired error probability and …
Non-binary low-density parity-check (LDPC) codes have some advantages over their binary counterparts, but unfortunately their decoding complexity is a significant challenge. The iterative hard- and soft-reliability based majority-logic decoding algorithms are attractive for non-binary LDPC codes, since they involve only finite field additions and multiplications as well as integer operations and hence have significantly lower complexity than other algorithms. In this paper, we propose two improvements to the majority-logic decoding algorithms. Instead of the accumulation of reliability informa…
We show that the binary Golay code is slightly more power efficient than the extended binary Golay code under maximum-likelihood (ML), hard-decision decoding. In fact, if a codeword from the extended code is transmitted, one cannot achieve a higher probability of correct decoding than by simply ignoring the 24th symbol and using an ML decoder for the non-extended code on the first 23 symbols. This is so, despite the fact that using that last symbol would allow one to sometimes correct error patterns with weight four. To our knowledge the worse performance of the extended Golay code has not bee…
The URLLC scenario in the upcoming 6G standard requires low latency and ultra reliable transmission, i.e., error correction towards ML performance. Achieving near-ML performance is very challenging especially for short block lengths. Polar codes are a promising candidate and already part of the 5G standard. The Successive Cancellation List (SCL) decoding algorithm provides very good error correction performance but at the cost of high computational decoding complexity resulting in large latency and low area and energy efficiency. Recently, Automorphism Ensemble Decoding (AED) gained a lot of a…
This dissertation focuses on fountain codes under maximum likelihood (ML) decoding. First LT codes are considered under a practical and widely used ML decoding algorithm known as inactivation decoding. Different analysis techniques are presented to characterize the decoding complexity. Next an upper bound to the probability of decoding failure of Raptor codes under ML decoding is provided. Then, the distance properties of an ensemble of fixed-rate Raptor codes with linear random outer codes are analyzed. Finally, a novel class of fountain codes is presented, which consists of a parallel concat…
Spatially coupled low-density parity-check codes show an outstanding performance under the low-complexity belief propagation (BP) decoding algorithm. They exhibit a peculiar convergence phenomenon above the BP threshold of the underlying non-coupled ensemble, with a wave-like convergence propagating through the spatial dimension of the graph, allowing to approach the MAP threshold. We focus on this particularly interesting regime in between the BP and MAP thresholds. On the binary erasure channel, it has been proved that the information propagates with a constant speed toward the successful de…
The serial concatenation of a repetition code with two or more accumulators has the advantage of a simple encoder structure. Furthermore, the resulting ensemble is asymptotically good and exhibits minimum distance growing linearly with block length. However, in practice these codes cannot be decoded by a maximum likelihood decoder, and iterative decoding schemes must be employed. For low-density parity-check codes, the notion of trapping sets has been introduced to estimate the performance of these codes under iterative message passing decoding. In this paper, we present a closed form finite l…
We propose a novel soft-aided iterative decoding algorithm for product codes (PCs). The proposed algorithm, named iterative bounded distance decoding with combined reliability (iBDD-CR), enhances the conventional iterative bounded distance decoding (iBDD) of PCs by exploiting some level of soft information. In particular, iBDD-CR can be seen as a modification of iBDD where the hard decisions of the row and column decoders are made based on a reliability estimate of the BDD outputs. The reliability estimates are derived using extrinsic message passing for generalized low-density-parity check (G…
A binary code is said to be a disjunctive list-decoding $s_L$-code, $s\ge1$, $L\ge1$, (briefly, LD $s_L$-code) if the code is identified by the incidence matrix of a family of finite sets in which the union of any $s$ sets can cover not more than $L-1$ other sets of the family. In this paper, we introduce a natural {\em probabilistic} generalization of LD $s_L$-code when the code is said to be an almost disjunctive LD $s_L$-code if the unions of {\em almost all} $s$ sets satisfy the given condition. We develop a random coding method based on the ensemble of binary constant-weight codes to obta…
A variety of low-density parity-check (LDPC) ensembles have now been observed to approach capacity with message-passing decoding. However, all of them use soft (i.e., non-binary) messages and a posteriori probability (APP) decoding of their component codes. In this paper, we show that one can approach capacity at high rates using iterative hard-decision decoding (HDD) of generalized product codes. Specifically, a class of spatially-coupled GLDPC codes with BCH component codes is considered, and it is observed that, in the high-rate regime, they can approach capacity under the proposed iterativ…
We propose a modified iterative bounded distance decoding of product codes. The proposed algorithm is based on exchanging hard messages iteratively and exploiting channel reliabilities to make hard decisions at each iteration. Performance improvements up to 0.26 dB are achieved.
