We introduce the notion of the stopping redundancy hierarchy of a linear block code as a measure of the trade-off between performance and complexity of iterative decoding for the binary erasure channel. We derive lower and upper bounds for the stopping redundancy hierarchy via Lovasz's Local Lemma and Bonferroni-type inequalities, and specialize them for codes with cyclic parity-check matrices. Based on the observed properties of parity-check matrices with good stopping redundancy characteristics, we develop a novel decoding technique, termed automorphism group decoding, that combines iterative message passing and permutation decoding. We also present bounds on the smallest number of permutations of an automorphism group decoder needed to correct any set of erasures up to a prescribed size. Simulation results demonstrate that for a large number of algebraic codes, the performance of the new decoding method is close to that of maximum likelihood decoding.
The $l$-th stopping redundancy $ρ_l(\mathcal C)$ of the binary $[n, k, d]$ code $\mathcal C$, $1 \le l \le d$, is defined as the minimum number of rows in the parity-check matrix of $\mathcal C$, such that the smallest stopping set is of size at least $l$. The stopping redundancy $ρ(\mathcal C)$ is defined as $ρ_d(\mathcal C)$. In this work, we improve on the probabilistic analysis of stopping redundancy, proposed by Han, Siegel and Vardy, which yields the best bounds known today. In our approach, we judiciously select the first few rows in the parity-check matrix, and then continue with the p…
Stopping sets play a crucial role in failure events of iterative decoders over a binary erasure channel (BEC). The $\ell$-th stopping redundancy is the minimum number of rows in the parity-check matrix of a code, which contains no stopping sets of size up to $\ell$. In this work, a notion of coverable stopping sets is defined. In order to achieve maximum-likelihood performance under iterative decoding over the BEC, the parity-check matrix should contain no coverable stopping sets of size $\ell$, for $1 \le \ell \le n-k$, where $n$ is the code length, $k$ is the code dimension. By estimating th…
We investigate the stopping redundancy hierarchy of linear block codes and its connection to permutation decoding techniques. An element in the ordered list of stopping redundancy values represents the smallest number of possibly linearly dependent rows in any parity-check matrix of a code that avoids stopping sets of a given size. Redundant parity-check equations can be shown to have a similar effect on decoding performance as permuting the coordinates of the received codeword according to a selected set of automorphisms of the code. Based on this finding we develop new decoding strategies fo…
It is now well known that the performance of a linear code $C$ under iterative decoding on a binary erasure channel (and other channels) is determined by the size of the smallest stopping set in the Tanner graph for $C$. Several recent papers refer to this parameter as the \emph{stopping distance} $s$ of $C$. This is somewhat of a misnomer since the size of the smallest stopping set in the Tanner graph for $C$ depends on the corresponding choice of a parity-check matrix. It is easy to see that $s \le d$, where $d$ is the minimum Hamming distance of $C$, and we show that it is always possible t…
Let C be a linear code with length n and minimum distance d. The stopping redundancy of C is defined as the minimum number of rows in a parity-check matrix for C such that the smallest stopping sets in the corresponding Tanner graph have size d. We derive new upper bounds on the stopping redundancy of linear codes in general, and of maximum distance separable (MDS) codes specifically, and show how they improve upon previously known results. For MDS codes, the new bounds are found by upper bounding the stopping redundancy by a combinatorial quantity closely related to Turan numbers. (The Turan …
For a linear block code C, its stopping redundancy is defined as the smallest number of check nodes in a Tanner graph for C, such that there exist no stopping sets of size smaller than the minimum distance of C. Schwartz and Vardy conjectured that the stopping redundancy of an MDS code should only depend on its length and minimum distance. We define the (n,t)-single-exclusion number, S(n,t) as the smallest number of t-subsets of an n-set, such that for each i-subset of the n-set, i=1,...,t+1, there exists a t-subset that contains all but one element of the i-subset. New upper bounds on the sin…
This paper is concerned with general analysis on the rank and row-redundancy of an array of circulants whose null space defines a QC-LDPC code. Based on the Fourier transform and the properties of conjugacy classes and Hadamard products of matrices, we derive tight upper bounds on rank and row-redundancy for general array of circulants, which make it possible to consider row-redundancy in constructions of QC-LDPC codes to achieve better performance. We further investigate the rank of two types of construction of QC-LDPC codes: constructions based on Vandermonde Matrices and Latin Squares and g…
In this letter, we propose an adaptive SC (Successive Cancellation)-List decoder for polar codes with CRC. This adaptive SC-List decoder iteratively increases the list size until the decoder outputs contain at least one survival path which can pass CRC. Simulation shows that the adaptive SC-List decoder provides significant complexity reduction. We also demonstrate that polar code (2048, 1024) with 24-bit CRC decoded by our proposed adaptive SC-List decoder with very large list size can achieve a frame error rate FER=0.001 at Eb/No=1.1dB, which is about 0.2dB from the information theoretic lim…
We generalize the notion of the stopping redundancy in order to study the smallest size of a trapping set in Tanner graphs of linear block codes. In this context, we introduce the notion of the trapping redundancy of a code, which quantifies the relationship between the number of redundant rows in any parity-check matrix of a given code and the size of its smallest trapping set. Trapping sets with certain parameter sizes are known to cause error-floors in the performance curves of iterative belief propagation decoders, and it is therefore important to identify decoding matrices that avoid such…
The AWGNC, BSC, and max-fractional pseudocodeword redundancy of a code is defined as the smallest number of rows in a parity-check matrix such that the corresponding minimum pseudoweight is equal to the minimum Hamming distance of the code. This paper provides new results on the AWGNC, BSC, and max-fractional pseudocodeword redundancies of codes. The pseudocodeword redundancies for all codes of small length (at most 9) are computed. Also, comprehensive results are provided on the cases of cyclic codes of length at most 250 for which the eigenvalue bound of Vontobel and Koetter is sharp.
The AWGNC, BSC, and max-fractional pseudocodeword redundancies of a binary linear code are defined to be the smallest number of rows in a parity-check matrix such that the corresponding minimum pseudoweight is equal to the minimum Hamming distance of the code. It is shown that most codes do not have a finite pseudocodeword redundancy. Also, upper bounds on the pseudocodeword redundancy for some families of codes, including codes based on designs, are provided. The pseudocodeword redundancies for all codes of small length (at most 9) are computed. Furthermore, comprehensive results are provided…
In this work, we study the minimum/stopping distance of array low-density parity-check (LDPC) codes. An array LDPC code is a quasi-cyclic LDPC code specified by two integers q and m, where q is an odd prime and m <= q. In the literature, the minimum/stopping distance of these codes (denoted by d(q,m) and h(q,m), respectively) has been thoroughly studied for m <= 5. Both exact results, for small values of q and m, and general (i.e., independent of q) bounds have been established. For m=6, the best known minimum distance upper bound, derived by Mittelholzer (IEEE Int. Symp. Inf. Theory, Jun./Jul…
Parameters of LDPC codes, such as minimum distance, stopping distance, stopping redundancy, girth of the Tanner graph, and their influence on the frame error rate performance of the BP, ML and near-ML decoding over a BEC and an AWGN channel are studied. Both random and structured LDPC codes are considered. In particular, the BP decoding is applied to the code parity-check matrices with an increasing number of redundant rows, and the convergence of the performance to that of the ML decoding is analyzed. A comparison of the simulated BP, ML, and near-ML performance with the improved theoretical …
This paper examines the construction of low-density parity-check (LDPC) codes from transversal designs based on sets of mutually orthogonal Latin squares (MOLS). By transferring the concept of configurations in combinatorial designs to the level of Latin squares, we thoroughly investigate the occurrence and avoidance of stopping sets for the arising codes. Stopping sets are known to determine the decoding performance over the binary erasure channel and should be avoided for small sizes. Based on large sets of simple-structured MOLS, we derive powerful constraints for the choice of suitable sub…
We study the application of polar codes in deletion channels by analyzing the cascade of a binary erasure channel (BEC) and a deletion channel. We show how polar codes can be used effectively on a BEC with a single deletion, and propose a list decoding algorithm with a cyclic redundancy check for this case. The decoding complexity is $O(N^2\log N)$, where $N$ is the blocklength of the code. An important contribution is an optimization of the amount of redundancy added to minimize the overall error probability. Our theoretical results are corroborated by numerical simulations which show that th…
We show that the performance of iterative belief propagation (BP) decoding of polar codes can be enhanced by decoding over different carefully chosen factor graph realizations. With a genie-aided stopping condition, it can achieve the successive cancellation list (SCL) decoding performance which has already been shown to achieve the maximum likelihood (ML) bound provided that the list size is sufficiently large. The proposed decoder is based on different realizations of the polar code factor graph with randomly permuted stages during decoding. Additionally, a different way of visualizing the p…
In a distributed storage environment, where the data is placed in nodes connected through a network, it is likely that one of these nodes fails. It is known that the use of erasure coding improves the fault tolerance and minimizes the redundancy added in distributed storage environments. The use of regenerating codes not only make the most of the erasure coding improvements, but also minimizes the amount of data needed to regenerate a failed node. In this paper, a new family of regenerating codes based on quasi-cyclic codes is presented. Quasi-cyclic flexible minimum storage regenerating (QCFM…
Polar codes were introduced in 2009 and proven to achieve the symmetric capacity of any binary-input discrete memoryless channel under low-complexity successive cancellation decoding. In this thesis, we construct cyclic polar codes based on a mixed-radix Cooley-Tukey decomposition of the Galois field Fourier transform. The main results are: we can, for the first time, construct, encode and decode polar codes that are cyclic, with their blocklength being arbitrary; for a given target block erasure rate, we can achieve significantly higher code rates on the erasure channel than the original pola…
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…
The 5th generation wireless systems (5G) standardization process of the 3rd generation partnership project (3GPP) chose polar codes as a channel coding scheme for the control channel. In case of downlink control information, polar codes are concatenated with distributed distributed cyclic redundancy check (CRC). Whereas CRC bits allow to improve the performance of successive cancellation list (SCL) decoders by improving distance properties, distributed CRC bits allow for path pruning and decoding early-termination. In this paper, we show how to take advantage of the distributed CRC to improve …
Permutation decoding gained recent interest as it can exploit the symmetries of a code in a parallel fashion. Moreover, it has been shown that by viewing permuted polar codes as polar subcodes, the set of usable permutations in permutation decoding can be increased. We extend this idea to pre-transformed polar codes, such as cyclic redundancy check (CRC)-aided polar codes, which previously could not be decoded using permutations due to their lack of automorphisms. Using belief propagation (BP)-based subdecoders, we showcase a performance close to CRC-aided SCL (CA-SCL) decoding. The proposed a…
Symbol-pair codes were proposed for the application in high density storage systems, where it is not possible to read individual symbols. Yaakobi, Bruck and Siegel proved that the minimum pair-distance of binary linear cyclic codes satisfies $d_2 \geq \lceil 3d_H/2 \rceil$ and introduced $b$-symbol metric codes in 2016. In this paper covering codes in $b$-symbol metrics are considered. Some examples are given to show that the Delsarte bound and the Norse bound for covering codes in the Hamming metric are not true for covering codes in the pair metric. We give the redundancy bound on covering r…
In this paper, we present list autoencoder (listAE) to mimic list decoding used in classical coding theory. With listAE, the decoder network outputs a list of decoded message word candidates. To train the listAE, a genie is assumed to be available at the output of the decoder. A specific loss function is proposed to optimize the performance of a genie-aided (GA) list decoding. The listAE is a general framework and can be used with any AE architecture. We propose a specific architecture, referred to as incremental-redundancy AE (IR-AE), which decodes the received word on a sequence of component…
Recently, rate-$1/ω$ zero-terminated and tail-biting convolutional codes (ZTCCs and TBCCs) with cyclic-redundancy-check (CRC)-aided list decoding have been shown to closely approach the random-coding union (RCU) bound for short blocklengths. This paper designs CRCs for rate-$(ω-1)/ω$ CCs with short blocklengths, considering both the ZT and TB cases. The CRC design seeks to optimize the frame error rate (FER) performance of the code resulting from the concatenation of the CRC and the CC. Utilization of the dual trellis proposed by Yamada \emph{et al.} lowers the complexity of CRC-aided serial l…
This paper is concerned with block Markov superposition transmission (BMST) of tail-biting convolutional code (TBCC). We propose a new decoding algorithm for BMST-TBCC, which integrates a serial list Viterbi algorithm (SLVA) with a soft check instead of conventional cyclic redundancy check (CRC). The basic idea is that, compared with an erroneous candidate codeword, the correct candidate codeword for the first sub-frame has less influence on the output of Viterbi algorithm for the second sub-frame. The threshold is then determined by statistical learning based on the introduced empirical diver…
In this work we propose an encoding and decoding framework for staircase codes based on non-systematic polar codes as component codes. The staircase structure allows for efficient parallelized decoding, while the polar component codes allow to benefit from the flexible structure and efficient soft-decision decoding algorithms. To enhance the performance of the polar staircase codes, we concatenate the polar component codes with cyclic redundancy check (CRC) outer codes, and we add interleavers within the staircase structure that are specific to polar code properties. The CRCs also allow to sub…
In this paper, a multi-antenna assisted virtual full-duplex (FD) relaying with reliability-aware iterative decoding at destination node is proposed to improve system spectral efficiency and reliability. This scheme enables two half-duplex relay nodes, mimicked as FD relaying, to alternatively serve as transmitter and receiver to relay their decoded data signals regardless the decoding errors, meanwhile, cancel the inter-relay interference with QR-decomposition. Then, by deploying the reliability-aware iterative detection/decoding process, destination node can efficiently mitigate inter-frame i…
For finite length polar codes, channel polarization leaves a significant number of channels not fully polarized. Adding a Cyclic Redundancy Check (CRC) to better protect information on the semi-polarized channels has already been successfully applied in the literature, and is straightforward to be used in combination with Successive Cancellation List (SCL) decoding. Belief Propagation (BP) decoding, however, offers more potential for exploiting parallelism in hardware implementation, and thus, we focus our attention on improving the BP decoder. Specifically, similar to the CRC strategy in the …
Polar codes are the first class of error correcting codes that provably achieve the channel capacity at infinite code length. They were selected for use in the fifth generation of cellular mobile communications (5G). In practical scenarios such as 5G, a cyclic redundancy check (CRC) is concatenated with polar codes to improve their finite length performance. This is mostly beneficial for sequential successive-cancellation list decoders. However, for parallel iterative belief propagation (BP) decoders, CRC is only used as an early stopping criterion with incremental error-correction performance…
Concatenating the state-of-the-art codes at moderate rates with repetition codes has emerged as a practical solution deployed in various standards for ultra-low-power devices such as in Internet-of-Things (IoT) networks. In this paper, we propose a novel concatenation mechanism for such applications which need to operate at very low signal-to-noise ratio (SNR) regime. In the proposed scheme, the outer code is a hybrid polar code constructed in two stages, one with a binary kernel and another also with a binary kernel but applied over a binary extension field. The inner code is a non-binary mul…
A deep-learning-aided successive-cancellation list (DL-SCL) decoding algorithm for polar codes is introduced with deep-learning-aided successive-cancellation (DL-SC) decoding being a specific case of it. The DL-SCL decoder works by allowing additional rounds of SCL decoding when the first SCL decoding attempt fails, using a novel bit-flipping metric. The proposed bit-flipping metric exploits the inherent relations between the information bits in polar codes that are represented by a correlation matrix. The correlation matrix is then optimized using emerging deep-learning techniques. Performanc…
Polar codes represent one of the major recent breakthroughs in coding theory and, because of their attractive features, they have been selected for the incoming 5G standard. As such, a lot of attention has been devoted to the development of decoding algorithms with good error performance and efficient hardware implementation. One of the leading candidates in this regard is represented by successive-cancellation list (SCL) decoding. However, its hardware implementation requires a large amount of memory. Recently, a partitioned SCL (PSCL) decoder has been proposed to significantly reduce the mem…
Although iterative decoding of polar codes has recently made huge progress based on the idea of permuted factor graphs, it still suffers from a non-negligible performance degradation when compared to state-of-the-art CRC-aided successive cancellation list (CA-SCL) decoding. In this work, we show that iterative decoding of polar codes based on the belief propagation list (BPL) algorithm can approach the error-rate performance of CA-SCL decoding and, thus, can be efficiently used for decoding the standardized 5G polar codes. Rather than only utilizing the cyclic redundancy check (CRC) as a stopp…
Cyclic redundancy check (CRC) aided polar codes are capable of achieving better performance than low-density parity-check (LDPC) codes under the successive cancelation list (SCL) decoding scheme. However, the SCL decoding scheme suffers from very high space and time complexities. Especially, the high space complexity is a major concern for adopting polar codes in modern mobile communication standards. In this paper, we propose a novel reduced-complexity successive cancelation list (R-SCL) decoding scheme which is effective to reduce the space complexity. Simulation results show that, with a (2…
Polar codes are a class of {\bf structured} channel codes proposed by Arıkan based on the principle of {\bf channel polarization}, and can {\bf achieve} the symmetric capacity of any Binary-input Discrete Memoryless Channel (B-DMC). The Soft CANcellation (SCAN) is a {\bf low-complexity} {\bf iterative} decoding algorithm of polar codes outperforming the widely-used Successive Cancellation (SC). Currently, in most cases, it is assumed that channel state is perfectly {\bf known} at the decoder and remains {\bf constant} during each codeword, which, however, is usually unrealistic. To decode pola…
Belief propagation (BP) is an iterative decoding algorithm for polar codes which can be parallelized effectively to achieve higher throughput. However, because of the presence of error floor due to cycles and stopping sets in the factor graph, the performance of the BP decoder is far from the performance of state of the art cyclic redundancy check (CRC) aided successive cancellation list (CA-SCL) decoders. It has been shown that successive BP decoding on multiple permuted factor graphs, which is called the multi-trellis BP decoder, can improve the error performance. However, when permuting the…
We consider the concatenation of a convolutional code (CC) with an optimized cyclic redundancy check (CRC) code as a promising paradigm for good short blocklength codes. The resulting CRC-aided convolutional code naturally permits the use of serial list Viterbi decoding (SLVD) to achieve maximum-likelihood decoding. The convolutional encoder of interest is of rate-$1/ω$ and the convolutional code is either zero-terminated (ZT) or tail-biting (TB). The resulting CRC-aided convolutional code is called a CRC-ZTCC or a CRC-TBCC. To design a good CRC-aided convolutional code, we propose the distanc…
We consider sparse superposition codes (SPARCs) over complex AWGN channels. Such codes can be efficiently decoded by an approximate message passing (AMP) decoder, whose performance can be predicted via so-called state evolution in the large-system limit. In this paper, we mainly focus on how to use concatenation of SPARCs and cyclic redundancy check (CRC) codes on the encoding side and use list decoding on the decoding side to improve the finite-length performance of the AMP decoder for SPARCs over complex AWGN channels. Simulation results show that such a concatenated coding scheme works much…
In successive cancellation (SC) polar decoding, an incorrect estimate of any prior unfrozen bit may bring about severe error propagation in the following decoding, thus it is desirable to find out and correct an error as early as possible. In this paper, we first construct a critical set $S$ of unfrozen bits, which with high probability (typically $>99\%$) includes the bit where the first error happens. Then we develop a progressive multi-level bit-flipping decoding algorithm to correct multiple errors over the multiple-layer critical sets each of which is constructed using the remaining undec…
Polar codes achieve outstanding error correction performance when using successive cancellation list (SCL) decoding with cyclic redundancy check. A larger list size brings better decoding performance and is essential for practical applications such as 5G communication networks. However, the decoding speed of SCL decreases with increased list size. Adaptive SCL (ASCL) decoding can greatly enhance the decoding speed, but the decoding latency for each codeword is different so A-SCL is not a good choice for hardware-based applications. In this paper, a hardware-friendly two-staged adaptive SCL (TA…
We present a novel iterative decoding algorithm for Reed-Muller (RM) codes, which takes advantage of a graph representation of the code. Vertices of the considered graph correspond to codewords, with two vertices being connected by an edge if and only if the Hamming distance between the corresponding codewords equals the minimum distance of the code. The algorithm uses a greedy local search to find a node optimizing a metric, e.g. the correlation between the received vector and the corresponding codeword. In addition, the cyclic redundancy check can be used to terminate the search as soon as a…
Long polar codes can achieve the capacity of arbitrary binary-input discrete memoryless channels under a low complexity successive cancelation (SC) decoding algorithm. But for polar codes with short and moderate code length, the decoding performance of the SC decoding algorithm is inferior. The cyclic redundancy check (CRC) aided successive cancelation list (SCL) decoding algorithm has better error performance than the SC decoding algorithm for short or moderate polar codes. However, the CRC aided SCL (CA-SCL) decoding algorithm still suffer from long decoding latency. In this paper, a reduced…
We present in this paper a special class of unit memory convolutional codes (UMCCs), called semi-random UMCCs (SRUMCCs), where the information block is first encoded by a short block code and then transmitted in a block Markov (random) superposition manner. We propose a successive cancellation list decoding algorithm, by which a list of candidate codewords are generated serially until one passes an empirical divergence test instead of the conventional cyclic redundancy check (CRC). The threshold for testing the correctness of candidate codewords can be learned off-line based on the statistical…
Low-rate and short-packet transmissions are important for ultra-reliable low-latency communications (URLLC). In this paper, we put forth a new family of sparse superposition codes for URLLC, called block orthogonal sparse superposition (BOSS) codes. We first present a code construction method for the efficient encoding of BOSS codes. The key idea is to construct codewords by the superposition of the orthogonal columns of a dictionary matrix with a sequential bit mapping strategy. We also propose an approximate maximum a posteriori probability (MAP) decoder with two stages. The approximate MAP …
Motivated by the need to communicate short control messages in 5G and beyond, this paper carefully designs codes for cyclic redundancy check (CRC)-aided list decoding of tail-biting convolutional codes (TBCCs) and polar codes. Both codes send a 32-bit message using an 11-bit CRC and 512 transmitted bits. We aim to provide a careful, fair comparison of the error performance and decoding complexity of polar and TBCC techniques for a specific case. Specifically, a TBCC is designed to match the rate of a (512, 43) polar code, and optimal 11-bit CRCs for both codes are designed. The paper examines …
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…
