Generalized concatenated codes were introduced in the 1970s by Zinoviev. There are many types of codes in the literature that are known by other names that can be viewed as generalized concatenated codes. Examples include matrix-product codes, multilevel codes and generalized cascade codes. Decoding algorithms for generalized concatenated codes were developed during the 1970s and 1980s. However, their use does not appear to be as widespread as it should, especially for codes that are known by other names but can be viewed as generalized concatenated codes.
In this paper we review the decoding algorithms for concatenated codes, generalized concatenated codes and matrix-product codes, and clarify the connection between matrix-product codes and generalized concatenated codes. We present a small improvement to the decoding algorithm for concatenated codes. We also extend the decoding algorithms from errors-only decoders to error-and-erasure decoders. Furthermore, we improve the upper bound on the computational complexity of the decoding algorithm in the case of matrix-product codes where the generator matrix for the inner code is non-singular by columns.
Concatenation is a method of building long codes out of shorter ones, it attempts to meet the problem of decoding complexity by breaking the required computation into manageable segments. Concatenated Continuous Phase Frequency Shift Keying (CPFSK) facilitates powerful error correction. CPFSK also has the advantage of being bandwidth efficient and compatible with nonlinear amplifiers. Bandwidth efficient concatenated coded modulation schemes were designed for communication over Additive White Gaussian noise (AWGN), and Rayleigh fading channels. An analytical bounds on the performance of serial…
Owing to their capacity-achieving performance and low encoding and decoding complexity, polar codes have drawn much research interests recently. Successive cancellation decoding (SCD) and belief propagation decoding (BPD) are two common approaches for decoding polar codes. SCD is sequential in nature while BPD can run in parallel. Thus BPD is more attractive for low latency applications. However BPD has some performance degradation at higher SNR when compared with SCD. Concatenating LDPC with Polar codes is one popular approach to enhance the performance of BPD , where a short LDPC code is use…
A scheme for concatenating the recently invented polar codes with interleaved block codes is considered. By concatenating binary polar codes with interleaved Reed-Solomon codes, we prove that the proposed concatenation scheme captures the capacity-achieving property of polar codes, while having a significantly better error-decay rate. We show that for any $ε> 0$, and total frame length $N$, the parameters of the scheme can be set such that the frame error probability is less than $2^{-N^{1-ε}}$, while the scheme is still capacity achieving. This improves upon $2^{-N^{0.5-\eps}}$, the frame err…
A scheme for concatenating the recently invented polar codes with non-binary MDS codes, as Reed-Solomon codes, is considered. By concatenating binary polar codes with interleaved Reed-Solomon codes, we prove that the proposed concatenation scheme captures the capacity-achieving property of polar codes, while having a significantly better error-decay rate. We show that for any $ε> 0$, and total frame length $N$, the parameters of the scheme can be set such that the frame error probability is less than $2^{-N^{1-ε}}$, while the scheme is still capacity achieving. This improves upon $2^{-N^{0.5-ε…
In this paper, we discuss pre-transformed RM-Polar codes and cyclic redundancy check (CRC) concatenated pre-transformed RM-Polar codes. The simulation results show that the pre-transformed RM-Polar (256, 128+9) code concatenated with 9-bit CRC can achieve a frame error rate (FER) of 0.001 at Eb/No=1.95dB under the list decoding, which is about 0.05dB from the RCU bound. The pre-transformed RM-Polar (512, 256+9) concatenated with 9-bit CRC can achieve a FER of 0.001 at Eb/No=1.6dB under the list decoding, which is 0.15dB from the RCU bound.
Decoding sequences that stem from multiple transmissions of a codeword over an insertion, deletion, and substitution channel is a critical component of efficient deoxyribonucleic acid (DNA) data storage systems. In this paper, we consider a concatenated coding scheme with an outer low-density parity-check code and either an inner convolutional code or a block code. We propose two new decoding algorithms for inference from multiple received sequences, both combining the inner code and channel to a joint hidden Markov model to infer symbolwise a posteriori probabilities (APPs). The first decoder…
Spatially coupled, parallel concatenated codes (SC-PCCs) have been shown to approach channel capacity when decoded using optimal iterative methods. However, under complexity constraints such decoding strategies can result in unacceptable power and latency costs. In this work, we employ convolutional self-orthogonal component codes along with low-complexity, suboptimal a posteriori probability (APP) threshold decoders with SC-PCCs to reduce decoding complexity. The proposed code design is faster, more energy efficient, and easier to implement than optimal methods, while offering significant cod…
An efficient scheme to increase the performance of polar codes at short and moderate block lengths is a concatenation of CRC code and a polar code. In order to obtain better result of the concatenation, a CRC code with best error control performance among all CRC codes with a fixed number of check bits has to be used. In this work we investigate CRC codes of 11 to 19 parity bits and determine those of them which have maximum minimum distance at any length it can be used. For CRC codes of 24 parity bits we were not able to perform complete search and we present the best obtained results. The in…
A new family of codes based on polar codes, soft concatenation and list+CRC decoding is proposed. Numerical experiments show the performance competitive with industry standards and Tal, Vardy approach.
We present a description of encoding/decoding for a concatenated quantum code that enables both protection against quantum computational errors and the occurrence of one quantum erasure. For this, it is presented how encoding and decoding for quantum graph codes are done, which will provide the protection against the occurrence of computational errors (external code). As internal code is used encoding and decoding via scheme of GHZ states for protection against the occurrence of one quantum erasure.
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…
Iterative decoding techniques have gain popularity due to their performance and their application in most communications systems. In this paper, we present a new application of our iterative decoder on the GPCB (Generalized Parallel Concatenated Block codes) which uses cyclic permutations. We introduce a new variant of the component decoder. After extensive simulation; the obtained result is very promising compared with several existing methods. We evaluate the effects of various parameters component codes, interleaver size, block size, and the number of iterations. Three interesting results a…
In this paper we provide a detailed description of Reed-Solomon (RS) codes, the most important algorithms for decoding them, and their use in concatenated coding systems for space applications. In the current literature there is scattered information regarding the bit-level implementation of such codes for either space systems or any other type of application. Consequently, we start with a general overview of the channel coding systems used in space communications and then we focus in the finest details. We first present a detailed description of the required algebra of RS codes with detailed …
Decoding sequences that stem from multiple transmissions of a codeword over an insertion, deletion, and substitution channel is a critical component of efficient deoxyribonucleic acid (DNA) data storage systems. In this paper, we consider a concatenated coding scheme with an outer nonbinary low-density parity-check code or a polar code and either an inner convolutional code or a time-varying block code. We propose two novel decoding algorithms for inference from multiple received sequences, both combining the inner code and channel to a joint hidden Markov model to infer symbolwise a posterior…
This paper considers iterative detection and decoding on the concatenated communication channel consisting of a two-dimensional magnetic recording (TDMR) channel modeled by the four-grain rectangular discrete grain model (DGM) proposed by Kavcic et. al., followed by a two-dimensional intersymbol interference (2D-ISI) channel modeled by linear convolution of the DGM model's output with a finite-extent 2D blurring mask followed by addition of white Gaussian noise. An iterative detection and decoding scheme combines TDMR detection, 2D-ISI detection, and soft-in/soft-out (SISO) channel decoding in…
We propose an alternative method for collaborative decoding of interleaved Reed-Solomon codes. Simulation results for a concatenated coding scheme using polar codes as inner codes are included.
A low-complexity soft-decision concatenated FEC scheme, consisting of an inner LDPC code and an outer staircase code is proposed. The inner code is tasked with reducing the bit error probability below the outer-code threshold. The concatenated code is obtained by optimizing the degree distribution of the inner-code ensemble to minimize estimated data-flow, for various choices of outer staircase codes. A key feature that emerges from this optimization is that it pays to leave some inner codeword bits completely uncoded, thereby greatly reducing a significant portion of the decoding complexity. …
It is known that the bit errors of polar codes with successive cancellation (SC) decoding are coupled. However, existing concatenation schemes of polar codes with other error correction codes rarely take this coupling effect into consideration. To achieve a better error performance of concatenated systems with polar codes as inner codes, one can divide all bits in an outer block into different polar blocks to completely de-correlate the possible coupled errors in the transmitter side. We call this interleaving a blind interleaving (BI) which serves as a benchmark. Two BI schemes, termed BI-DP …
Generalized Concatenated (GC), also known as Integrated Interleaved (II) Codes, are studied from an erasure correction point of view making them useful for Redundant Arrays of Independent Disks (RAID) types of architectures combining global and local properties. The fundamental erasure-correcting properties of the codes are proven and efficient encoding and decoding algorithms are provided. Although less powerful than the recently developed PMDS codes, this implementation has the advantage of allowing generalization to any range of parameters while the size of the field is much smaller than th…
We introduce generalized spatially coupled parallel concatenated codes (GSC-PCCs), a class of spatially coupled turbo-like codes obtained by coupling parallel concatenated codes (PCCs) with a fraction of information bits repeated before the PCC encoding. GSC-PCCs can be seen as a generalization of the original spatially coupled parallel concatenated convolutional codes (SC-PCCs) proposed by Moloudi et al. [1]. To characterize the asymptotic performance of GSC-PCCs, we derive the corresponding density evolution equations and compute their decoding thresholds. We show that the proposed codes hav…
Two concatenated coding schemes based on fixed-rate Raptor codes are proposed for error control in NAND flash memory. One is geared for off-line recovery of uncorrectable pages and the other is designed for page error correction during the normal read mode. Both proposed coding strategies assume hard-decision decoding of the inner code with inner decoding failure generating erasure symbols for the outer Raptor code. Raptor codes allow low-complexity decoding of very long codewords while providing capacity- approaching performance for erasure channels. For the off-line page recovery scheme, one…
We investigate threshold-based multi-trial decoding of concatenated codes with an inner Maximum-Likelihood decoder and an outer error/erasure (L+1)/L-extended Bounded Distance decoder, i.e. a decoder which corrects e errors and t erasures if e(L+1)/L + t <= d - 1, where d is the minimum distance of the outer code and L is a positive integer. This is a generalization of Forney's GMD decoding, which was considered only for L = 1, i.e. outer Bounded Minimum Distance decoding. One important example for (L+1)/L-extended Bounded Distance decoders is decoding of L-Interleaved Reed-Solomon codes. Our …
Unsourced random access (URA) has emerged as a pragmatic framework for next-generation distributed sensor networks. Within URA, concatenated coding structures are often employed to ensure that the central base station can accurately recover the set of sent codewords during a given transmission period. Many URA algorithms employ independent inner and outer decoders, which can help reduce computational complexity at the expense of a decay in performance. In this article, an enhanced decoding algorithm is presented for a concatenated coding structure consisting of a wide range of inner codes and …
In this work, we give good concatenated code ensembles for the binary erasure channel (BEC). In particular, we consider repeat multiple-accumulate (RMA) code ensembles formed by the serial concatenation of a repetition code with multiple accumulators, and the hybrid concatenated code (HCC) ensembles recently introduced by Koller et al. (5th Int. Symp. on Turbo Codes & Rel. Topics, Lausanne, Switzerland) consisting of an outer multiple parallel concatenated code serially concatenated with an inner accumulator. We introduce stopping sets for iterative constituent code oriented decoding using max…
An alternative method for collaborative decoding of interleaved Reed-Solomon codes as well as Gabidulin codes for the case of high interleaving degree is proposed. As an example of application, simulation results are presented for a concatenated coding scheme using polar codes as inner codes.
Polar codes were introduced by Arikan in 2008 and are the first family of error-correcting codes achieving the symmetric capacity of an arbitrary binary-input discrete memoryless channel under low complexity encoding and using an efficient successive cancellation decoding strategy. Recently, non-binary polar codes have been studied, in which one can use different algebraic geometry codes to achieve better error decoding probability. In this paper, we study the performance of binary polar codes that are obtained from non-binary algebraic geometry codes using concatenation. For binary polar code…
In this paper, the analysis of the performance of the concatenation of a short polar code with an outer binary linear block code is addressed from a distance spectrum viewpoint. The analysis targets the case where an outer cyclic code is employed together with an inner systematic polar code. A concatenated code ensemble is introduced placing an interleaver at the input of the polar encoder. The introduced ensemble allows deriving bounds on the achievable error rates under maximum likelihood decoding, by applying the union bound to the (expurgated) average weight enumerators. The analysis sugge…
This work provides an algebraic framework for source coding with decoder side information and its dual problem, channel coding with encoder side information, showing that nested concatenated codes can achieve the corresponding rate-distortion and capacity-noise bounds. We show that code concatenation preserves the nested properties of codes and that only one of the concatenated codes needs to be nested, which opens up a wide range of possible new code combinations for these side information based problems. In particular, the practically important binary version of these problems can be address…
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…
A new class of spatially-coupled turbo-like codes (SC-TCs), dubbed generalized spatially coupled parallel concatenated codes (GSC-PCCs), is introduced. These codes are constructed by applying spatial coupling on parallel concatenated codes (PCCs) with a fraction of information bits repeated $q$ times. GSC-PCCs can be seen as a generalization of the original spatially-coupled parallel concatenated codes proposed by Moloudi et al. [2]. To characterize the asymptotic performance of GSC-PCCs, we derive the corresponding density evolution equations and compute their decoding thresholds. The thresho…
This paper deals with the application of list decoding of Reed--Solomon codes to a concatenated code for key reproduction using Physical Unclonable Functions. The resulting codes achieve a higher error-correction performance at the same code rate than known schemes in this scenario. We also show that their decoding algorithms can be protected from side-channel attacks on the runtime both by masking techniques and by directly modifying the algorithms to have constant runtime.
The errors occurring in DNA-based storage are correlated in nature, which is a direct consequence of the synthesis and sequencing processes. In this paper, we consider the memory-$k$ nanopore channel model recently introduced by Hamoum et al., which models the inherent memory of the channel. We derive the maximum a posteriori (MAP) decoder for this channel model. The derived MAP decoder allows us to compute achievable information rates for the true DNA storage channel assuming a mismatched decoder matched to the memory-$k$ nanopore channel model, and quantify the loss in performance assuming a…
A successive cancellation list (SCL) decoder with limited list size for polar codes can not be analyzed as a successive cancellation (SC) decoder, nor as a maximum likelihood (ML) decoder, due to the complicated decoding errors caused by path elimination. To address this issue, an analytical tool, named as cluster pairwise error probability (CPEP), is proposed in this paper to measure the competitiveness of the correct path against the error paths in an SCL decoder. It is shown that the sum of CPEPs over error paths could be used as an indicator of the probability of correct path being elimina…
DNA Data storage has recently attracted much attention due to its durable preservation and extremely high information density (bits per gram) properties. In this work, we propose a hybrid coding strategy comprising of generalized constrained codes to tackle homopolymer (run-length) limit and a protograph based low-density parity-check (LDPC) code to correct asymmetric nucleotide level (i.e., A/T/C/G) substitution errors that may occur in the process of DNA sequencing. Two sequencing techniques namely, Nanopore sequencer and Illumina sequencer with their equivalent channel models and capacities…
Polar codes have been shown to approach capacity of symmetric binary erasure channels and also have low encoding and decoding complexity. Wireless channels are bursty in nature and a method to apply polar codes over wireless channels is proposed. This paper investigates the performance of polar codes over typical urban fading wireless channels. The wireless channel is modeled as a degraded erasure channel by using an inner convolution code and interleaving. When the convolution code fails to correct any errors, the corresponding bits are discarded and treated as erasures. We present the perfor…
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 …
This paper investigates properties of polar codes that can be potentially useful in real-world applications. We start with analyzing the performance of finite-length polar codes over the binary erasure channel (BEC), while assuming belief propagation as the decoding method. We provide a stopping set analysis for the factor graph of polar codes, where we find the size of the minimum stopping set. We also find the girth of the graph for polar codes. Our analysis along with bit error rate (BER) simulations demonstrate that finite-length polar codes show superior error floor performance compared t…
The security of public-key cryptosystems is mostly based on number theoretic problems like factorization and the discrete logarithm. There exists an algorithm which solves these problems in polynomial time using a quantum computer. Hence, these cryptosystems will be broken as soon as quantum computers emerge. Code-based cryptography is an alternative which resists quantum computers since its security is based on an NP-complete problem, namely decoding of random linear codes. The McEliece cryptosystem is the most prominent scheme to realize code-based cryptography. Many codeclasses were propose…
By inserting interleavers between intermediate stages of the polar encoder, a new class of polar codes, termed interleaved polar (i-polar) codes, is proposed. By the uniform interleaver assumption, we derive the weight enumerating function (WEF) and input-output weight enumerating function (IOWEF) averaged over the ensemble of i-polar codes. The average WEF can be used to calculate the upper bound on the average block error rate (BLER) of a code selected at random from the ensemble of i-polar codes. Also, we propose a concatenated coding scheme that employs P high rate codes as the outer code …
Spatially coupled serially concatenated codes (SC-SCCs) are a class of spatially coupled turbo-like codes, which have a close-to-capacity performance and low error floor. In this paper we investigate the impact of coupling memory, block length, decoding window size, and number of iterations on the performance, complexity, and latency of SC-SCCs. Several design tradeoffs are presented to see the relation between these parameters in a wide range. Also, our analysis provides design guidelines for SC-SCCs in different scenarios to make the code design independent of block length. As a result, bloc…
Undetected errors are important for linear codes, which are the only type of errors after hard decision and automatic-repeat-request (ARQ), but do not receive much attention on their correction. In concatenated channel coding, suboptimal source coding and joint source-channel coding, constrains among successive codewords may be utilized to improve decoding performance. In this paper, list decoding is used to correct the undetected errors. The benefit proportion of the correction is obviously improved especially on Hamming codes and Reed-Muller codes, which achieves about 40% in some cases. But…
A novel fountain coding scheme has been introduced. The scheme consists of a parallel concatenation of a MDS block code with a LRFC code, both constructed over the same field, $F_q$. The performance of the concatenated fountain coding scheme has been analyzed through derivation of tight bounds on the probability of decoding failure as a function of the overhead. It has been shown how the concatenated scheme performs as well as LRFC codes in channels characterized by high erasure probabilities, whereas they provide failure probabilities lower by several orders of magnitude at moderate/low erasu…
In this paper, we present a new construction of asymmetric quantum codes (AQCs) by combining classical concatenated codes (CCs) with tensor product codes (TPCs), called asymmetric quantum concatenated and tensor product codes (AQCTPCs) which have the following three advantages. First, only the outer codes in AQCTPCs need to satisfy the orthogonal constraint in quantum codes, and any classical linear code can be used for the inner, which makes AQCTPCs very easy to construct. Second, most AQCTPCs are highly degenerate, which means they can correct many more errors than their classical TPC counte…
Turbo codes and CRC codes are usually decoded separately according to the serially concatenated inner codes and outer codes respectively. In this letter, we propose a hybrid decoding algorithm of turbo-CRC codes, where the outer codes, CRC codes, are not used for error detection but as an assistance to improve the error correction performance. Two independent iterative decoding and reliability based decoding are carried out in a hybrid schedule, which can efficiently decode the two different codes as an entire codeword. By introducing an efficient error detecting method based on normalized Euc…
