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Braided convolutional codes (BCCs) are a type of parallel-concatenated convolutional code in which the parity outputs of one component encoder are fed back and used as inputs to the other component encoder at the succeeding time unit. In this paper, we review the published results on BCCs that have appeared in the literature over the past ten years and present a unified view of BCCs in the context of other types of turbo-like codes. We also include some recent results on iterative decoding thresholds for BCCs, weight enumerators and distance growth rates, window decoding for low latency operation, and rate-compatible BCCs for high-rate applications.

In this paper, we present a random puncturing analysis of low-density parity-check (LDPC) code ensembles. We derive a simple analytic expression for the iterative belief propagation (BP) decoding threshold of a randomly punctured LDPC code ensemble on the binary erasure channel (BEC) and show that, with respect to the BP threshold, the strength and suitability of an LDPC code ensemble for random puncturing is completely determined by a single constant that depends only on the rate and the BP threshold of the mother code ensemble. We then provide an efficient way to accurately predict BP thresholds of randomly punctured LDPC code ensembles on the binary- input additive white Gaussian noise channel (BI-AWGNC), given only the BP threshold of the mother code ensemble on the BEC and the design rate, and we show how the prediction can be improved with knowledge of the BI-AWGNC threshold. We also perform an asymptotic minimum distance analysis of randomly punctured code ensembles and present simulation results that confirm the robust decoding performance promised by the asymptotic results. Protograph-based LDPC block code and spatially coupled LDPC code ensembles are used throughout as examples to demonstrate the results.

In this paper, we study the block error rate (BLER) performance of spatially coupled low-density parity-check (SC- LDPC) codes using a sliding window decoder suited for streaming applications. Previous studies of SC-LDPC have focused on the bit error rate (BER) performance or the frame error rate (FER) performance over the entire length of the code. Here, we consider protograph-based constructions of SC-LDPC codes in which a window decoder continuously outputs blocks in a streaming fashion, and we examine the BLER associated with these blocks.We begin by examining the effect of protograph design on the streaming BLER by varying the block size and the coupling width in such a way that the overall constraint length of the SC-LDPC code remains constant. Next, we investigate the BLER scaling behavior with block size and coupling width. Lastly, we consider the effect of employing an outer code to protect blocks, so that small numbers of residual errors can be corrected by the outer code. Simulation results for the additive white Gaussian noise channel (AWGNC) are included and comparisons are made to LDPC block codes (LDPC-BCs).

In this paper, we construct protograph-based spatially coupled low-density parity-check (LDPC) codes by coupling together a series of L disjoint, or uncoupled, LDPC code Tanner graphs into a single coupled chain. By varying L, we obtain a flexible family of code ensembles with varying rates and frame lengths that can share the same encoding and decoding architecture for arbitrary L. We demonstrate that the resulting codes combine the best features of optimized irregular and regular codes in one design: capacity approaching iterative belief propagation (BP) decoding thresholds and linear growth of minimum distance with block length. In particular, we show that, for sufficiently large L, the BP thresholds on both the binary erasure channel and the binary-input additive white Gaussian noise channel saturate to a particular value significantly better than the BP decoding threshold and numerically indistinguishable from the optimal maximum a posteriori decoding threshold of the uncoupled LDPC code. When all variable nodes in the coupled chain have degree greater than two, asymptotically the error probability converges at least doubly exponentially with decoding iterations and we obtain sequences of asymptotically good LDPC codes with fast convergence rates and BP thresholds close to the Shannon limit. Further, the gap to capacity decreases as the density of the graph increases, opening up a new way to construct capacity achieving codes on memoryless binary-input symmetric-output channels with low-complexity BP decoding.

We consider a window decoding scheme for Braided Convolutional Codes (BCCs) based on the BCJR algorithm. We describe the principle of this decoding scheme and show that BCCs with window decoding exhibit excellent performance. The tradeoff between performance and decoding latency is examined and, to reduce decoding complexity, both uniform and nonuniform message passing schedules within the decoding window are considered. Finally, puncturing is employed to obtain rate-compatible code rates of 1/2 and 2/3 starting from a rate 1/3 example mother code. Simulation results show that, with nonuniform message passing and puncturing, near capacity performance can be maintained throughout the rate range considered with reasonable decoding complexity and no visible error floors.

Spatially-coupled regular LDPC code ensembles have outstanding performance with belief propagation decoding and can perform arbitrarily close to the Shannon limit without requiring irregular graph structures. In this paper, we are concerned with the performance and complexity of spatially-coupled ensembles with a rate-compatibility constraint. Spatially-coupled regular ensembles that support rate-compatibility through extension have been proposed before and show very good performance if the node degrees and the coupling width are chosen appropriately. But due to the strict constraint of maintaining a regular degree, there exist certain unfavorable rates that exhibit bad performance and high decoding complexity. We introduce an altered LDPC ensemble construction that changes the evolution of degrees over subsequent incremental redundancy steps in such a way, that the degrees can be kept low to achieve outstanding performance close to Shannon limit for all rates. These ensembles always outperform their regular counterparts at small coupling width.

Spatially-Coupled LDPC (SC-LDPC) codes have been recently shown to be very efficient for transmissions over nonergodic channels, in particular over block-fading channels [1]. In fact, it is possible to design a SC-LDPC code with any given code diversity [2]. In this work, we investigate the performance of SC-LDPC codes over block-fading channels, assuming a mismatch (or offset) between the first bit of a transmission packet and a first bit of a codeword. Such a mismatch is called the synchronisation offset, and it has a negative impact on the code diversity. We propose a data-allocation scheme for SC-LDPC codes that allows to obtain a robustness to the synchronisation offset. Combined together with the code design from [2], it allows to design efficient SC-LDPC codes, whose performance degrades only slowly under imperfect transmission conditions.

This work focuses on the design of SC-LDPC codes for transmission over non-ergodic, block-fading channels. Our main contribution is an algorithm, allowing to start from a (J,K )-regular, uncoupled LDPC ensemble, from which one can recursively build up a protograph-based SC-LDPC ensemble having any target diversity order d. The diversity order is achieved assuming a low-complexity iterative decoding algorithm. The increase of d comes at the cost of increasing the memory constraint (i.e., the coupling parameter) of the SC-LDPC ensemble.

In this paper, we provide an efficient way to predict iterative belief propagation (BP) decoding thresholds of randomly punctured low-density parity-check (LDPC) code ensembles on the binary-input additive white Gaussian noise channel (AWGNC), given only the BP threshold of the mother code ensemble on the binary erasure channel (BEC) and the code design rate. We show that the predictions are accurate by comparing them with values calculated by discretized density evolution for a variety of puncturing fractions. We find that the strength and suitability of an LDPC code ensemble for random puncturing over the AWGNC with respect to iterative decoding threshold is completely determined by a single constant, and this behavior is demonstrated using both LDPC block code and spatially coupled LDPC code ensembles. Finally, we present simulation results that confirm the excellent decoding performance promised by the asymptotic results.

Deliverable D11.2 del projecte europeu NEWCOM#, The report presents the Intermediate Results of N# JRAs on Performance Limits of Wireless Communications and highlights the fundamental issues that have been investigated by the WP1.1. The report illustrates the Joint Research Activities (JRAs) already identified during the first year of the project which are currently ongoing. For each activity there is a description, an illustration of the adherence and relevance with the identified fundamental open issues, a short presentation of the preliminary results, and a roadmap for the joint research work in the next year. Appendices for each JRA give technical details on the scientific activity in each JRA., Peer Reviewed, Preprint

Spatially coupled regular LDPC code ensembles have outstanding performance with belief propagation decoding and can perform close to the Shannon limit. In this paper we investigate the suitability of coupled regular LDPC code ensembles with respect to rate-flexibility. Regular ensembles with good performance and low complexity exist for a variety of specific code rates. On the other hand it can be observed that outside this set of favorable rational rates the complexity and performance become unreasonably high. We therefore propose ensembles with slight irregularity that allow us to smoothly cover the complete range of rational rates. Our simple construction allows a performance with negligible gap to the Shannon limit while maintaining complexity as low as for the best regular code ensembles. At the same time the construction guarantees that asymptotically the minimum distance grows linearly with the length of the coupled blocks.

In this paper, we give an overview of spatially coupled turbo codes (SC-TCs), the spatial coupling of parallel and serially concatenated convolutional codes, recently introduced by the authors. For presentation purposes, we focus on spatially coupled serially concatenated codes (SC-SCCs). We review the main principles of SC-TCs and discuss their exact density evolution (DE) analysis on the binary erasure channel. We also consider the construction of a family of rate-compatible SC-SCCs with simple 4-state component encoders. For all considered code rates, threshold saturation of the belief propagation (BP) to the maximum a posteriori threshold of the uncoupled ensemble is demonstrated, and it is shown that the BP threshold approaches the Shannon limit as the coupling memory increases. Finally we give some simulation results for finite lengths.

In this paper, we study random puncturing of protograph-based spatially coupled low-density parity-check (SC- LDPC) code ensembles. We show that, with respect to iterative decoding threshold, the strength and suitability of an LDPC code ensemble for random puncturing over the binary erasure channel (BEC) is completely determined by a single constant that depends only on the rate and iterative decoding threshold of the mother code ensemble. We then use this analysis to show that randomly punctured SC-LDPC code ensembles display near capacity thresholds for a wide range of rates. We also perform an asymptotic minimum distance analysis and show that, like the SC-LDPC mother code ensemble, the punctured SC-LDPC code ensembles are also asymptotically good. Finally, we present some simulation results that confirm the excellent decoding performance promised by the asymptotic results.

Spatially coupled low-density parity-check (SC-LDPC) codes are considered for transmission over the block-fading channel. The diversity order of the SC-LDPC codes is studied using density evolution and simulation results. We demonstrate that the diversity order of the code can be increased, without lowering the code rate, by simply increasing the coupling parameter (memory) of a SC-LDPC code. For a (3,6)-regular SC-LDPC code with rate R = 1/2 and memory mcc = 4 a remarkable diversity of d = 10 is achieved without the need for any specific code structure. The memory of the SC-LDPC codes makes them robust against a non-stationary mobile-radio environment. The decoding of SC-LDPC codes using a latency constrained sliding window decoder is also considered.

Braided convolutional codes (BCCs) are a class of spatially coupled turbo-like codes with a structure that is similar to product codes or generalized LDPC codes. We derive explicit input/output transfer functions of the component convolutional decoders for the binary erasure channel (BEC). These are then used to formulate exact density evolution equations for blockwise BCCs under belief propagation (BP) decoding with optimal component APP decoders. Thresholds are computed for the coupled and uncoupled case, which is equivalent to tailbiting. Due to the relatively high rate of the component codes a significant threshold improvement by spatial coupling can be observed.

Window decoding schedules are very attractive for message passing decoding of spatially coupled LDPC codes. They take advantage of the inherent convolutional code structure and allow continuous transmission with low decoding latency and complexity. In this paper we show that the decoding complexity can be further reduced if suitable message passing schedules are applied within the decoding window. An improvement based schedule is presented that easily adapts to different ensemble structures, window sizes, and channel parameters. Its combination with a serial (on-demand) schedule is also considered. Results from a computer search based schedule are shown for comparison.

We compare LDPC block and LDPC convolutional codes with respect to their decoding performance under low decoding latencies. Protograph based regular LDPC codes are considered with rather small lifting factors. LDPC block and convolutional codes are decoded using belief propagation. For LDPC convolutional codes, a sliding window decoder with different window sizes is applied to continuously decode the input symbols. We show the required Eb/N0 to achieve a bit error rate of 10 -5 for the LDPC block and LDPC convolutional codes for the decoding latency of up to approximately 550 information bits. It has been observed that LDPC convolutional codes perform better than the block codes from which they are derived even at low latency. We demonstrate the trade off between complexity and performance in terms of lifting factor and window size for a fixed value of latency. Furthermore, the two codes are also compared in terms of their complexity as a function of Eb/N0. Convolutional codes with Viterbi decoding are also compared with the two above mentioned codes.

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 decoding thresholds, ensembles of GSC-LDPC codes are asymptotically good and have large minimum distance growth rates.