Outer Code Research Articles

Polar Coded Repetition

Constructing efficient low-rate error-correcting codes with low-complexity encoding and decoding has become increasingly important for applications involving ultra-low-power devices such as Internet-of-Things (IoT). To this end, schemes based on concatenating the state-of-the-art codes at moderate rates with repetition codes have emerged as practical solutions deployed in various standards. In this paper, we propose a novel mechanism for concatenating outer polar codes with inner repetition codes which we refer to as polar coded repetition. More specifically, we propose to transmit a slightly modified polar codeword by deviating from Arıkan’s standard <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2 \times 2$ </tex-math></inline-formula> Kernel in a certain number of polarization recursions at each repetition block. We show how this modification can improve the asymptotic achievable rate of the standard polar-repetition scheme, while ensuring that the overall encoding and decoding complexity is kept almost the same. The achievable rate is analyzed for the binary erasure channel (BEC) and additive white Gaussian noise (AWGN) channel. Moreover, we show that the finite-length performance of the polar coded repetition scheme under cyclic redundancy check (CRC) aided successive cancellation list (SCL) decoder over AWGN channel is better than the uncoded polar-repetition scheme at the cost of a slight increase in decoding complexity. We also compare the proposed scheme, in terms of performance and complexity, with other low-rate solution based on polar codes in the literature.

Hybrid Non-Binary Repeated Polar Codes

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 multiplicative repetition code. This particular structure inherits low-complexity decoding structures of polar codes while enabling concatenation with an inner non-binary multiplicative repetition scheme. The decoding for the proposed scheme is done using cyclic redundancy check (CRC) aided successive cancellation list (SCL) decoder over additive white Gaussian noise (AWGN) and Rayleigh fading channels. Simulation results demonstrate that the proposed hybrid non-binary repeated polar code provides performance gain compared to a polar-repetition scheme with comparable decoding complexity.

Generalized Spatially-Coupled Parallel Concatenated Codes With Partial Repetition

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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> times. GSC-PCCs can be seen as a generalization of the original spatially-coupled parallel concatenated codes proposed by Moloudi <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al.</i> , 2017. To characterize the asymptotic performance of GSC-PCCs, we derive the corresponding density evolution equations and compute their decoding thresholds. The threshold saturation effect is observed and proven. Most importantly, we rigorously prove that the rate- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$R$ </tex-math></inline-formula> GSC-PCC ensemble with 2-state convolutional component codes achieves at least a fraction <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$1-\frac {R}{R+q}$ </tex-math></inline-formula> of the capacity of the binary erasure channel (BEC) for repetition factor <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q\geq 2$ </tex-math></inline-formula> and this multiplicative gap vanishes as <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> tends to infinity. To the best of our knowledge, this is the first class of SC-TCs that are proven to be capacity-achieving. Further, the connection between the strength of the component codes, the decoding thresholds of GSC-PCCs, and the repetition factor is established. The superiority of the proposed codes with finite blocklength is exemplified by comparing their error performance with that of existing SC-TCs via computer simulations.

CRC-Aided Adaptive BP Decoding of PAC Codes.

Although long polar codes with successive cancellation decoding can asymptotically achieve channel capacity, the performance of short blocklength polar codes is far from optimal. Recently, Arıkan proposed employing a convolutional pre-transformation before the polarization network, called polarization-adjusted convolutional (PAC) codes. In this paper, we focus on improving the performance of short PAC codes concatenated with a cyclic redundancy check (CRC) outer code, CRC-PAC codes, since error detection capability is essential in practical applications, such as the polar coding scheme for the control channel. We propose an enhanced adaptive belief propagation (ABP) decoding algorithm with the assistance of CRC bits for PAC codes. We also derive joint parity-check matrices of CRC-PAC codes suitable for iterative BP decoding. The proposed CRC-aided ABP (CA-ABP) decoding can effectively improve error performance when partial CRC bits are used in the decoding. Meanwhile, the error detection ability can still be guaranteed by the remaining CRC bits and adaptive decoding parameters. Moreover, compared with the conventional CRC-aided list (CA-List) decoding, our proposed scheme can significantly reduce computational complexity, to achieve a better trade-off between the performance and complexity for short PAC codes.

CAMPUR KODE DAN ALIH KODE DALAM LIRIK LAGU “SENANDUNG RINDU” SYUBBANUL MUSLIMIN PERSPEKTIF SOSIOLINGUISTIK

This research about code mixing and code switching in sociolinguistic studies. The analysis of code mixing and code switching was carried out throught the lyrics of the song “Senandung Rindu” by the Syubbanul Muslimin, rebana vocal group. The background of code code mixing and code switching in the lyrics of the song because Indonesian people have bilingual or multilingual. They are able to use two or more different languages when interacting wich each other. This research was conducted with a descriptive qualitative research method. The method of data collection by recording, free listening, recording, and data recapitulation techniques. The result of this research are : there are eight code mixing that found in the lyrics of the song “Senandung Rindu”, the form is outer code mixing and also of alternation, that found one form of clause code mixing and the other of a phrase, there are six code switching that found in the song’s lyrics and the form is internal code switching.

Finite Rate QLDPC-GKP Coding Scheme that Surpasses the CSS Hamming Bound

Quantum error correction has recently been shown to benefit greatly from specific physical encodings of the code qubits. In particular, several researchers have considered the individual code qubits being encoded with the continuous variable GottesmanKitaev-Preskill (GKP) code, and then imposed an outer discrete-variable code such as the surface code on these GKP qubits. Under such a concatenation scheme, the analog information from the inner GKP error correction improves the noise threshold of the outer code. However, the surface code has vanishing rate and demands a lot of resources with growing distance. In this work, we concatenate the GKP code with generic quantum low-density parity-check (QLDPC) codes and demonstrate a natural way to exploit the GKP analog information in iterative decoding algorithms. We first show the noise thresholds for two lifted product QLDPC code families, and then show the improvements of noise thresholds when the iterative decoder – a hardware-friendly min-sum algorithm (MSA) – utilizes the GKP analog information. We also show that, when the GKP analog information is combined with a sequential update schedule for MSA, the scheme surpasses the well-known CSS Hamming bound for these code families. Furthermore, we observe that the GKP analog information helps the iterative decoder in escaping harmful trapping sets in the Tanner graph of the QLDPC code, thereby eliminating or significantly lowering the error floor of the logical error rate curves. Finally, we discuss new fundamental and practical questions that arise from this work on channel capacity under GKP analog information, and on improving decoder design and analysis.

A concatenated LDPC-marker code for channels with correlated insertion and deletion errors in bit-patterned media recording system.

Most synchronization error correction codes deal with random independent insertion and deletion errors without correlation. In this paper, we propose a probabilistic channel model with correlated insertion and deletion (CID) errors to capture the data dependence applicable to the bit-patterned media recording (BPMR) system. We also investigate the error performance and decoding complexity of a concatenated LDPC-marker code over the CID channel. Furthermore, we modify the forward backward decoding algorithm to make it suitable for the CID channel, and elaborate it based on a two-dimensional state transition diagram. Compared with the conventional marker coding scheme dealing with random errors, the concatenated LDPC-marker code takes into account the dependence between synchronization errors, improves the error performance, and reduces the decoding complexity. The BER performance of the concatenated LDPC-marker code is improved by more than 50% on average, and the decoding time is reduced by nearly 35% when the LDPC code (n = 4521, k = 3552) and the marker code (Nm = 2, Nc = 30) are used over the CID channel.

Analysis of PAM Mapping for Convolutional-Coded Physical-Layer Network Coding

In this letter, we investigate iterative decoding for convolutional-coded physical-layer network coding operated with bit-interleaved coded modulation (BICM-PNC) over two-way relay channel (TWRC). The superimposed constellation at the relay from two users may not remain the same characteristic as the original modulation at the users. Thus, we first propose a design of mapping for PAM modulations at the users and enable the superimposed constellation of the maximum squared Euclidean weight (MSEW). Then, we analyze the PNC mappings by means of an extrinsic information transfer (EXIT) chart. The analysis and simulation results show that the proposed mapping can provide a good match to a given outer channel code. In particular, it achieves much better error performance than the conventional semi-Gray and binary-bit mappings at high SNR region for both 8-PAM and 16-PAM modulations.

Quantum Lego: Building Quantum Error Correction Codes from Tensor Networks

We introduce a flexible and graphically intuitive framework that constructs complex quantum error correction codes from simple codes or states, generalizing code concatenation. More specifically, we represent the complex code constructions as tensor networks built from the tensors of simple codes or states in a modular fashion. Using a set of local moves known as operator pushing, one can derive properties of the more complex codes, such as transversal non-Clifford gates, by tracing the flow of operators in the network. The framework endows a network geometry to any code it builds and is valid for constructing stabilizer codes as well as non-stabilizer codes over qubits and qudits. For a contractible tensor network, the sequence of contractions also constructs a decoding/encoding circuit. To highlight the framework's range of capabilities and to provide a tutorial, we lay out some examples where we glue together simple stabilizer codes to construct non-trivial codes. These examples include the toric code and its variants, a holographic code with transversal non-Clifford operators, a 3d stabilizer code, and other stabilizer codes with interesting properties. Surprisingly, we find that the surface code is equivalent to the 2d Bacon-Shor code after "dualizing" its tensor network encoding map.

Rate-Adaptive Concatenated Multi-Level Coding With Novel Probabilistic Amplitude Shaping

This paper proposes a new probabilistic amplitude shaping (PAS) approach for concatenated two-level multi-level coding (MLC). The proposed system is based on a concatenated forward error correction (FEC) scheme where outer codes are serially concatenated with inner two-level MLC. This concatenated two-level MLC scheme has recently been shown to have a potential for achieving better performance-complexity trade-offs than the conventional bit-interleaved coded modulation (BICM). Meanwhile, PAS has recently been demonstrated to offer remarkable performance gains as well as rate adaptivity. However, the majority of existing works on PAS assume the use of the binary reflected Gray code as a bit-labeling, and its application to coded modulation schemes with other bit-labelings, such as two-level MLC, may not be straightforward. In this paper, we devise a bit-labeling scheme and propose a new PAS structure for an efficient integration of PAS with two-level MLC systems. More specifically, we propose to generate <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">signed</i> amplitude symbols with the distribution matcher (DM) for maximizing both coding and shaping gains achieved by two-level MLC and PAS, respectively, while the conventional PAS generates <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">unsigned</i> amplitude symbols. It is demonstrated by simulation results that, with 256QAM and inner polar codes, the proposed two-level MLC with PAS simultaneously offers 75% reduction in the number of required inner encoding and soft-decision (SD) decoding operations for given outer and inner FEC code lengths, and up to 0.3 dB performance gain over the conventional PAS scheme.

Massive Unsourced Random Access: Exploiting Angular Domain Sparsity

This paper investigates the unsourced random access (URA) scheme to accommodate numerous machine-type users communicating to a base station equipped with multiple antennas. Existing works adopt a slotted transmission strategy to reduce system complexity; they operate under the framework of coupled compressed sensing (CCS) which concatenates an outer tree code to an inner compressed sensing code for slot-wise message stitching. We suggest that by exploiting the MIMO channel information in the angular domain, redundancies required by the tree encoder/decoder in CCS can be removed to improve spectral efficiency, thereby an uncoupled transmission protocol is devised. To perform activity detection and channel estimation, we propose an expectation-maximization-aided generalized approximate message passing algorithm with a Markov random field support structure, which captures the inherent clustered sparsity structure of the angular domain channel. Then, message reconstruction in the form of a clustering decoder is performed by recognizing slot-distributed channels of each active user based on similarity. We put forward the slot-balanced <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$ K $ </tex-math></inline-formula> -means algorithm as the kernel of the clustering decoder, resolving constraints and collisions specific to the application scene. Extensive simulations reveal that the proposed scheme achieves a better error performance at high spectral efficiency compared to the CCS-based URA schemes.

Unsourced Random Access With Coded Compressed Sensing: Integrating AMP and Belief Propagation

Sparse regression codes with approximate message passing (AMP) decoding have gained much attention in recent times. The concepts underlying this coding scheme extend to unsourced random access with coded compressed sensing (CCS), as first demonstrated by Fengler, Jung, and Caire. Specifically, their approach employs a concatenated coding framework with an inner AMP decoder followed by an outer tree decoder. In their original implementation, these two components work independently of each other, with the tree decoder acting on the static output of the AMP decoder. This article introduces a novel framework where the inner AMP decoder and the outer decoder operate in tandem, dynamically passing information back and forth to take full advantage of the underlying CCS structure. This scheme necessitates the redesign of the outer code as to enable belief propagation in a computationally tractable manner. The enhanced architecture exhibits significant performance benefits over a range of system parameters. The error performance of the proposed scheme can be accurately predicted through a set of equations known as state evolution of AMP. These findings are supported both analytically and through numerical methods.

Inter-Multilevel Super-Orthogonal Space-Time Coding Scheme for Reliable ZigBee-Based IoMT Communications.

The Internet of Things (IoT) technology has revolutionized the healthcare industry by enabling a new paradigm for healthcare delivery. This paradigm is known as the Internet of Medical Things (IoMT). IoMT devices are typically connected via a wide range of wireless communication technologies, such as Bluetooth, radio-frequency identification (RFID), ZigBee, Wi-Fi, and cellular networks. The ZigBee protocol is considered to be an ideal protocol for IoMT communication due to its low cost, low power usage, easy implementation, and appropriate level of security. However, maintaining ZigBee’s high reliability is a major challenge due to multi-path fading and interference from coexisting wireless networks. This has increased the demand for more efficient channel coding schemes that can achieve a more reliable transmission of vital patient data for ZigBee-based IoMT communications. To meet this demand, a novel coding scheme called inter-multilevel super-orthogonal space–time coding (IM-SOSTC) can be implemented by combining the multilevel coding and set partitioning of super-orthogonal space–time block codes based on the coding gain distance (CGD) criterion. The proposed IM-SOSTC utilizes a technique that provides inter-level dependency between adjacent multilevel coded blocks to facilitate high spectral efficiency, which has been compromised previously by the high coding gain due to the multilevel outer code. In this paper, the performance of IM-SOSTC is compared to other related schemes via a computer simulation that utilizes the quasi-static Rayleigh fading channel. The simulation results show that IM-SOSTC outperforms other related coding schemes and is capable of providing the optimal trade-off between coding gain and spectral efficiency whilst guaranteeing full diversity and low complexity.

PAPR reduction methods based on bit flipping

ABSTRACT In this work, two peak-to-average-power ratio (PAPR) reduction methods based on bit flipping are designed for OFDM signals. Rather than sending side information, error correction codes are used to recover the flipped bits, as well as to protect the transmitted data. The first PAPR reduction method combines hard-decision decoding (HDD) and natural mapping. Taking advantage of the fact that the natural mapping symbol is represented as a linear combination of the included bits, the flipping bits can be selected at low complexity. The concept of compressed sensing is introduced in order to minimize the number of flipping bits and retain most of the code distance for error correction. In the second method, a low-density parity-check (LDPC) code is cascaded as an inner code following the HDD code, and Gray mapping is used. Soft-decision decoding (SDD) is applied to the LDPC code, which significantly improves the data protection capability, then the outer HDD code is able to focus on recovering the flipped bits. The designs proposed in this paper show great flexibility and provide a trade-off between PAPR reduction and the error-correction capability.

Hybrid neural coded modulation: Design and training methods

We propose a hybrid coded modulation scheme which composes of inner and outer codes. The outer-code can be any standard binary linear code with efficient soft decoding capability (e.g. low-density parity-check (LDPC) codes). The inner code is designed using a deep neural network (DNN) which takes the channel coded bits and outputs modulated symbols. For training the DNN, we propose to use a loss function that is inspired by the generalized mutual information. The resulting constellations are shown to outperform the conventional quadrature amplitude modulation (QAM) based coding scheme for modulation order 16 and 64 with 5G standard LDPC codes.

Code Constructions and Bounds for Identification via Channels

Consider the identification (ID) via channels problem, where a receiver decides whether the transmitted identifier is its identifier, rather than decoding it. This model allows to transmit identifiers whose size scales doubly-exponentially in the blocklength, unlike common transmission codes with exponential scaling. Binary constant-weight codes (CWCs) suffice to achieve the ID capacity. Relating parameters of a binary CWC to the minimum distance of a code and using higher-order correlation moments, two upper bounds on binary CWC sizes are proposed. These bounds are also upper bounds on identifier sizes for ID codes constructed by using binary CWCs. We propose two constructions based on optical orthogonal codes (OOCs), which are used in optical multiple access schemes, have constant-weight codewords, and satisfy cyclic cross-correlation and auto-correlation constraints. These constructions are modified and concatenated with outer Reed-Solomon codes to propose new binary CWCs being optimal for ID. Improvements to the finite-parameter performance are shown by using outer codes with larger minimum distance vs. blocklength ratios. We illustrate ID regimes for which our ID code constructions perform significantly better than existing constructions.

Doppler Shift Tolerance of Accumulation and Outer Coding in MIMO-PMCW Radar

Phase-modulated continuous wave (PMCW) has been widely regarded as a promising modulation scheme for radar systems, e.g., in highly automated driving (HAD) applications. Although the so-called outer coding can efficiently enable the multiple-input-multiple-output (MIMO) operation of PMCW-based radar systems, the yielded processing gain in this multiplexing approach may be reduced at increasing Doppler shifts. In this context, this letter introduces a normalized Doppler shift parameter that enables predicting the Doppler-shift-induced degradation of the processing gain in a MIMO-PMCW radar system. Finally, simulation and measurement results confirm the usefulness of the introduced parameter in desigining MIMO-PMCW radars.

Parity-Check Coding Transmit Diversity for Wireless Communications With High Mobility

Highmobility is an important scenario for the 5G wireless communications. However, the high mobility results in serious time-varying fading, bringing challenges to the reliabilities of wireless transmission. Diversity technique is an essential method to overcome the effect of fading. It is noted that an LDPC (low-density parity-check) code is a concatenation of single parity-check codes (SPC) and repetition codes (REP) by a random interleaver, and a REP can be viewed as a typical example for achieving time/frequency diversity. Thus, it is appealing to jointly consider LDPC codes and diversity techniques. This paper presents parity-check coding transmit diversity (PCTD) schemes based on half-rate invertible LDPC codes, which are capable of achieving both diversity and coding gains from the time-varying fading channels. Besides, we investigate two types of PCTD schemes, and take into consideration two transmission modes, which are single-carrier (SC) and orthogonal frequency division multiplexing (OFDM) transmission modes. A chase combining based message passing algorithm (CC-MPA) is proposed for the proposed schemes, which can take full advantages of signal processing and coding gains. Furthermore, the Shannon limits of time-varying fading channels with BPSK signaling are deduced. Simulation results show that the proposed PCTD with CC-MPA algorithm can provide better BER performances than the classical time-domain diversity scheme in time-varying fading channels, which is appealing for the high mobility communications.

An Enhanced Decoding Algorithm for Coded Compressed Sensing with Applications to Unsourced Random Access.

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 an outer tree-based code. It is shown that this algorithmic enhancement has the potential to simultaneously improve error performance and decrease the computational complexity of the decoder. This enhanced decoding algorithm is applied to two existing URA algorithms, and the performance benefits of the algorithm are characterized. Findings are supported by numerical simulations.

FSW: Fulcrum Sliding Window Coding for Low-Latency Communication

Fulcrum Random Linear Network Coding (RLNC) combines outer coding in a large Galois Field, e.g., GF(28), with inner coding in GF(2) to flexibly trade off the strong protection (low probability of linear dependent coding coefficients) of GF(28) with the low computational complexity of GF(2). However, the existing Fulcrum RLNC approaches are generation based, leading to large packet delays due to the joint processing of all packets in a generation in the encoder and decoder. In order to avoid these delays, we introduce Fulcrum Sliding Window (FSW) coding. We introduce two flavors of FSW: Fulcrum Nonsystematic Sliding Window (FNSW), which divides a given generation into multiple partially overlapping blocks, and Fulcrum Systematic Sliding Window (FSSW), which intersperses coded packets among the uncoded (systematic) transmission of the source packets in a generation. Our extensive evaluations indicate that FSSW substantially reduces the in-order packet delay (for moderately large generation and window sizes down to less than one fourth) and more than doubles the encoding and decoding (computation) throughput compared to generation-based Fulcrum.