An ensemble of LDPC convolutional codes with parity-check matrices composed of permutation matrices is considered. The convergence of the iterative belief propagation based decoder for terminated convolutional codes in the ensemble is analyzed for binary-input output-symmetric memoryless channels using density evolution techniques. We observe that the structured irregularity in the Tanner graph of the codes leads to significantly better thresholds when compared to corresponding LDPC block codes.
We analyze the finite-length performance of spatially coupled low-density parity-check (SC-LDPC) codes under window decoding over the binary erasure channel. In particular, we propose a refinement of the scaling law by Olmos and Urbanke for the frame error rate (FER) of terminated SC-LDPC ensembles under full belief propagation (BP) decoding. The refined scaling law models the decoding process as two independent Ornstein-Uhlenbeck processes, in correspondence to the two decoding waves that propagate toward the center of the coupled chain for terminated SC-LDPC codes. We then extend the propose…
The minimum pseudoweight is an important parameter related to the decoding performance of LDPC codes with iterative message-passing decoding. In this paper, we consider ensembles of periodically time-varying spatially coupled LDPC (SC-LDPC) codes and the pseudocodewords arising from their finite graph covers of a fixed degree. We show that for certain $(J,K)$-regular SC-LDPC code ensembles and a fixed cover degree, the typical minimum pseudoweight of the unterminated (and associated tail-biting/terminated) SC-LDPC code ensembles grows linearly with the constraint (block) length as the constrai…
Spatially coupled low-density parity-check (SC-LDPC) codes can achieve the channel capacity under low-complexity belief propagation (BP) decoding, however, there is a non-negligible rate-loss because of termination effects for practical finite coupling lengths. In this paper, we study how we can approach the performance of terminated SC-LDPC codes by random shortening of tail-biting SC-LDPC codes. We find the minimum required rate-loss in order to achieve the same performance than terminated codes. We additionally study the use of tail-biting SC-LDPC codes for transmission over parallel channe…
This paper investigates the design and performance of delayed bit-interleaved coded modulation (DBICM) with low-density parity-check (LDPC) codes. For Gray labeled square $M$-ary quadrature amplitude modulation (QAM) constellations, we investigate the optimal delay scheme with the largest spectrum efficiency of DBICM for a fixed maximum number of delayed time slots and a given signal-to-noise ratio. When analyzing the capacity of DBICM, we find two important properties: the capacity improvement due to delayed coded bits being mapped to the real and imaginary parts of the transmitted symbols ar…
We show how the iterative decoding threshold of tailbiting spatially coupled (SC) low-density parity-check (LDPC) code ensembles can be improved over the binary input additive white Gaussian noise channel by allowing the use of different transmission energies for the codeword bits. We refer to the proposed approach as energy shaping. We focus on the special case where the transmission energy of a bit is selected among two values, and where a contiguous portion of the codeword is transmitted with the largest one. Given these constraints, an optimal energy boosting policy is derived by means of …
This paper is concerned with the design of capacity approaching ensembles of Low-Densiy Parity-Check (LDPC) codes for correlated sources. We consider correlated binary sources where the data is encoded independently at each source through a systematic LDPC encoder and sent over two independent channels. At the receiver, a iterative joint decoder consisting of two component LDPC decoders is considered where the encoded bits at the output of each component decoder are used at the other decoder as the a priori information. We first provide asymptotic performance analysis using the concept of extr…
We propose a novel encoding/transmission scheme called continuous chain (CC) transmission that is able to improve the finite-length performance of a system using spatially-coupled low-density parity-check (SC-LDPC) codes. In CC transmission, instead of transmitting a sequence of independent codewords from a terminated SC-LDPC code chain, we connect multiple chains in a layered format, where encoding, transmission, and decoding are now performed in a continuous fashion. The connections between chains are created at specific points, chosen to improve the finite-length performance of the code str…
For finite coupling lengths, terminated spatially coupled low-density parity-check (SC-LDPC) codes show a non-negligible rate-loss. In this paper, we investigate if this rate loss can be mitigated by tail-biting SC-LDPC codes in conjunction with iterative demapping of higher order modulation formats. Therefore, we examine the BP threshold of different coupled and uncoupled ensembles. A comparison between the decoding thresholds approximated by EXIT charts and the density evolution results of the coupled and uncoupled ensemble is given. We investigate the effect and potential of different label…
Protograph low-density-parity-check (LDPC) are considered to design near-capacity low-rate codes over the binary erasure channel (BEC) and binary additive white Gaussian noise (BIAWGN) channel. For protographs with degree-one variable nodes and doubly-generalized LDPC (DGLDPC) codes, conditions are derived to ensure equality of bit-error threshold and block-error threshold. Using this condition, low-rate codes with block-error threshold close to capacity are designed and shown to have better error rate performance than other existing codes.
This paper presents a theoretical study of a new type of LDPC codes motivated by practical storage applications. LDPCL codes (suffix L represents locality) are LDPC codes that can be decoded either as usual over the full code block, or locally when a smaller sub-block is accessed (to reduce latency). LDPCL codes are designed to maximize the error-correction performance vs. rate in the usual (global) mode, while at the same time providing a certain performance in the local mode. We develop a theoretical framework for the design of LDPCL codes. Our results include a design tool to construct an L…
In this paper, we formulate a new problem to cope with the transmission of extra bits over an existing coded transmission link (referred to as coded payload link) without any cost of extra transmission energy or extra bandwidth. This is possible since a gap to the channel capacity typically exists for a practical code. A new concept, termed as accessible capacity, is introduced to specify the maximum rate at which the superposition transmission of extra bits is reliable and has a negligible effect on the performance of the coded payload link. For a binary-input output-symmetric (BIOS) memoryle…
Belief propagation applied to iterative decoding and sparse recovery through approximate message passing (AMP) are two research areas that have seen monumental progress in recent decades. Inspired by these advances, this article introduces sparse regression LDPC codes and their decoding. Sparse regression codes (SPARCs) are a class of error correcting codes that build on ideas from compressed sensing and can be decoded using AMP. In certain settings, SPARCs are known to achieve capacity; yet, their performance suffers at finite block lengths. Likewise, LDPC codes can be decoded efficiently usi…
In this paper, we present a novel way for solving the main problem of designing the capacity approaching irregular low-density parity-check (LDPC) code ensemble over binary erasure channel (BEC). The proposed method is much simpler, faster, accurate and practical than other methods. Our method does not use any relaxation or any approximate solution like previous works. Our method works and finds optimal answer for any given check node degree distribution. The proposed method was implemented and it works well in practice with polynomial time complexity. As a result, we represent some degree dis…
This paper considers the forward error correction (FEC) code design for approaching the capacity of a dynamic multiple access channel (MAC) where both the number of users and their respective signal powers keep constantly changing, resembling the scenario of an actual wireless cellular system. To obtain a low-complexity non-orthogonal multiple access (NOMA) scheme, we propose a serial concatenation of a low-density parity-check (LDPC) code and a repetition code (REP), this way achieving near Gaussian MAC (GMAC) capacity performance while coping with the dynamics of the MAC system. The joint op…
Spatially-Coupled LDPC (SC-LDPC) ensembles achieve the capacity of binary memoryless channels (BMS), asymptotically, under belief-propagation (BP) decoding. In this paper, we study the BP decoding of these code ensembles over a BMS channel and in the presence of a single random burst of erasures. We show that in the limit of code length, codewords can be recovered successfully if the length of the burst is smaller than some maximum recoverable burst length. We observe that the maximum recoverable burst length is practically the same if the transmission takes place over binary erasure channel o…
In this work, the design of robust, protograph-based low-density parity-check (LDPC) codes for rate-adaptive communication via probabilistic shaping is considered. Recently, probabilistic amplitude shaping (PAS) by Böcherer et al. has been introduced for capacity approaching and rate-adaptive communication with a bitwise-demapper and binary decoder. Previous work by the authors considered the optimization of protograph based LDPC codes for PAS and specific spectral efficiencies (SEs) to jointly optimize the LDPC code node degrees and the mapping of the coded bits to the bit-interleaved coded m…
Continuous-variable quantum key distribution utilizes an ensemble of coherent states of light to distribute secret encryption keys between two parties. One of the challenges is thereby the requirement of capacity approaching error correcting codes in the low signal-to-noise (SNR) regime (SNR < 0 dB). Multilevel coding (MLC) combined with multistage decoding (MSD) can solve this challenge in combination with multi-edge-type low-density parity-check (MET-LDPC) codes which are ideal for low code rates in the low SNR regime due to degree-one variable nodes. However, the complexity of designing suc…
Spatially coupled low-density parity-check (SC-LDPC) codes are sparse graph codes that have recently become of interest due to their capacity-approaching performance on memoryless binary input channels. In this paper, we unify all existing SC-LDPC code construction methods under a new generalized description of SC-LDPC codes based on algebraic lifts of graphs. We present an improved low-complexity counting method for the special case of $(3,3)$-absorbing sets for array-based SC-LDPC codes, which we then use to optimize permutation assignments in SC-LDPC code construction. We show that codes co…
In this chapter, we show how the use of differential coding and the presence of phase slips in the transmission channel affect the total achievable information rates and capacity of a system. By means of the commonly used QPSK modulation, we show that the use of differential coding does not decrease the total amount of reliably conveyable information over the channel. It is a common misconception that the use of differential coding introduces an unavoidable differential loss. This perceived differential loss is rather a consequence of simplified differential detection and decoding at the recei…
The joint design of input constellation and low-density parity-check (LDPC) codes to approach the symmetric capacity of the two-user Gaussian multiple access channel is studied. More specifically, multilevel coding is employed at each user to construct a high-order input constellation and the constellations of the users are jointly designed so as to maximize the multiuser shaping gain. At the receiver, each layer of the multilevel coding is jointly decoded among users, while successive cancellation is employed across layers. The LDPC code employed by each user in each layer is designed using E…
