The Euclidean projection operation is the most complex and time-consuming of the alternating direction method of multipliers (ADMM) decoding algorithms, resulting in a large number of resources when deployed on hardware platforms. We propose a simplified line segment projection algorithm (SLSA) and present the hardware design and the quantization scheme of the SLSA. In simulation results, the proposed SLSA module has a better performance than the original algorithm with the same fixed bitwidths due to the centrosymmetric structure of SLSA. Furthermore, the proposed SLSA module with a simpler structure without hypercube projection can reduce time consuming by up to 72.2% and reduce hardware resource usage by more than 87% compared to other Euclidean projection modules in the experiments.

In this letter, we study low-density parity-check (LDPC) codes for noisy channels with insertion and deletion (ID) errors. We first propose a design method of irregular LDPC codes for such channels, which can be used to simultaneously obtain degree distributions for different noise levels. We then show the asymptotic/finite-length decoding performances of designed codes and compare them with the symmetric information rates of cascaded ID-noisy channels. Moreover, we examine the relationship between decoding performance and a code structure of irregular LDPC codes.

We propose a coding/decoding strategy that surpasses the symmetric information rate of a binary insertion/deletion (ID) channel and approaches the Markov capacity of the channel. The proposed codes comprise inner trellis codes and outer irregular low-density parity-check (LDPC) codes. The trellis codes are designed to mimic the transition probabilities of a Markov input process that achieves a high information rate, whereas the LDPC codes are designed to maximize an iterative decoding threshold in the superchannel (concatenation of the ID channels and trellis codes).

In this letter, the structural analysis of nonbinary cyclic and quasi-cyclic (QC) low-density parity-check (LDPC) codes with α-multiplied parity-check matrices (PCMs) is concerned. Using analytical methods, several structural parameters of nonbinary cyclic and QC LDPC codes with α-multiplied PCMs are determined. In particular, some classes of nonbinary LDPC codes constructed from finite fields and finite geometries are shown to have good minimum and stopping distances properties, which may explain to some extent their wonderful decoding performances.

Over the past two decades, irregular low-density parity-check (LDPC) codes have not been able to decode information corrupted by insertion and deletion (ID) errors without markers. In this paper, we bring to light the existence of irregular LDPC codes that approach the symmetric information rates (SIR) of the channel with ID errors, even without markers. These codes have peculiar shapes in their check-node degree distributions. Specifically, the check-node degrees are scattered and there are degree-2 check nodes. We propose a code construction method based on the progressive edge-growth algorithm tailored for the scattered check-node degree distributions, which enables the SIR-approaching codes to progress in the finite-length regime. Moreover, the SIR-approaching codes demonstrate asymptotic and finite-length performance that outperform the existing counterparts, namely, concatenated coding of irregular LDPC codes with markers and spatially coupled LDPC codes.

We propose hardware-aware sum-product (SP) decoding for low-density parity-check codes. To simplify an implementation using a fixed-point number representation, we transform SP decoding in the logarithm domain to that in the decision domain. A polynomial approximation is proposed to implement an update rule of the proposed SP decoding efficiently. Numerical simulations show that the approximate SP decoding achieves almost the same performance as the exact SP decoding when an appropriate degree in the polynomial approximation is used, that it improves the convergence properties of SP and normalized min-sum decoding in the high signal-to-noise ratio regime, and that it is robust against quantization errors.

The separating redundancy is an important concept in the analysis of the error-and-erasure decoding of a linear block code using a parity-check matrix of the code. In this letter, we derive new constructive upper bounds on the second separating redundancies of low-density parity-check (LDPC) codes constructed from projective and Euclidean planes over the field Fq with q even.

Based on the codewords of the [q,2,q-1] extended Reed-Solomon (RS) code over the finite field Fq, we can construct a regular binary γq×q2 matrix H(γ,q), where q is a power of 2 and γ≤q. The matrix H(γ,q) defines a regular low-density parity-check (LDPC) code C(γ,q), called a full-length RS-LDPC code. Using some analytical methods, we completely determine the values of s(H(4,q)), s(H(5,q)), and d(C(5,q)) in this letter, where s(H(γ,q)) and d(C(γ,q)) are the stopping distance of H(γ,q) and the minimum distance of C(γ,q), respectively.

In this letter, we present a new scheme to find small fundamental instantons (SFIs) of regular low-density parity-check (LDPC) codes for the linear programming (LP) decoding over the binary symmetric channel (BSC). Based on the fact that each instanton-induced graph (IIG) contains at least one short cycle, we determine potential instantons by constructing possible IIGs which contain short cycles and additional paths connected to the cycles. Then we identify actual instantons from potential ones under the LP decoding. Simulation results on some typical LDPC codes show that our scheme is effective, and more instantons can be obtained by the proposed scheme when compared with the existing instanton search method.

In this paper, an area-efficient decoder architecture is proposed for the quasi-cyclic low-density parity check (QC-LDPC) codes specified in the IEEE 802.16e WiMAX standard. The decoder supports all the code rates and codeword lengths defined in the standard. In order to achieve low area and maximize hardware utilization, the decoder utilizes 4 decoding function units, which is the greatest common divisor of the expansion factors. In addition, the decoder adopts a novel scheduling scheme named stride scheduling, which stores the extrinsic messages in non-sequential order to replace the conventional complex flexible permutation network with simple small-sized cyclic shifters and also minimize the number of memory accesses. To further minimize the complexity, the number of extrinsic memory instances for 24 block columns is reduced to 5 banks by identifying independent sets. All the memory instances used in the decoder are single-port memories which cost less area and price compared to dual-port ones. Finally, the decoding function units have partially parallel structure to make the decoding throughput sufficiently over the requirement of the WiMAX standard. The proposed decoder is synthesized with 49 K equivalent gates and 54,144 bits of memory, and the implementation occupies 0.40 mm2 in a 65 nm CMOS technology.

In this paper, through extrinsic information transfer (EXIT) band chart analysis, an adaptive iterative decoding approach (AIDA) is proposed to reduce the iterative decoding complexity and delay for finite-length differentially encoded Low-density parity-check (DE-LDPC) coded systems with multiple-symbol differential detection (MSDD). The proposed AIDA can adaptively adjust the observation window size (OWS) of the MSDD soft-input soft-output demodulator (SISOD) and the outer iteration number of the iterative decoder (consisting of the MSDD SISOD and the LDPC decoder) instead of setting fixed values for the two parameters of the considered systems. The performance of AIDA depends on its stopping criterion (SC) which is used to terminate the iterative decoding before reaching the maximum outer iteration number. Many SCs have been proposed; however, these approaches focus on turbo coded systems, and it has been proven that they do not well suit for LDPC coded systems. To solve this problem, a new SC called differential mutual information (DMI) criterion, which can track the convergence status of the iterative decoding, is proposed; it is based on tracking the difference of the output mutual information of the LDPC decoder between two consecutive outer iterations of the considered systems. AIDA using the DMI criterion can adaptively adjust the out iteration number and OWS according to the convergence situation of the iterative decoding. Simulation results show that compared with using the existing SCs, AIDA using the DMI criterion can further reduce the decoding complexity and delay, and its performance is not affected by a change in the LDPC code and transmission channel parameters.

A simplified algorithm for check node processing of extended min-sum (EMS) q-ary LDPC decoders is presented in this letter. Compared with the bubble check algorithm, the so-called dynamic bubble-check (DBC) algorithm aims to further reduce the computational complexity for the elementary check node (ECN) processing. By introducing two flag vectors in ECN processing, The DBC algorithm can use the minimum number of comparisons at each step. Simulation results show that, DBC algorithm uses significantly fewer comparison operations than the bubble check algorithm, and presents no performance loss compared with standard EMS algorithm on AWGN channels.

Given an odd prime q and an integer m ≤ q, an array-based parity-check matrix H(m,q) can be constructed for a quasi-cyclic low-density parity-check (LDPC) code C(m,q). For m=4 and q ≥ 11, we prove the stopping distance of H(4,q) is 10, which is equal to the minimum Hamming distance of the associated code C(4,q). In addition, a tighter lower bound on the stopping distance of H(m,q) is also given for m > 4 and q ≥ 11.

Quasi-cyclic (QC) low-density parity-check (LDPC) codes have several appealing properties regarding decoding, storage requirements and encoding aspects. In this paper, we focus on the QC LDPC codes over GF(q) whose parity-check matrices have fixed column weight j = 2. By investigating two subgraphs in the Tanner graphs of the corresponding base matrices, we derive two upper bounds on the minimum Hamming distance for this class of codes. In addition, a method is proposed to construct QC LDPC codes over GF(q), which have good Hamming distance distributions. Simulations show that our designed codes have good performance.

Partially parallel decoding architectures are widely used in the design of low-density parity-check (LDPC) decoders, especially for quasi-cyclic (QC) LDPC codes. To comply with the code structure of parity-check matrices of QC-LDPC codes, many small memory blocks are conventionally employed in this architecture. The total memory area usually dominates the area requirement of LDPC decoders. This paper proposes a low-complexity memory access architecture that merges small memory blocks into memory groups to relax the effect of peripherals in small memory blocks. A simple but efficient algorithm is also presented to handle the additional delay elements introduced in the memory merging method. Experiment results on a rate-1/2 parity-check matrix defined in the IEEE 802.16e standard show that the LDPC decoder designed using the proposed memory access architecture has the lowest area complexity among related studies. Compared to a design with the same specifications, the decoder implemented using the proposed architecture requires 33% fewer gates and is more power-efficient. The proposed new memory access architecture is thus suitable for the design of low-complexity LDPC decoders.

Irregular low-density parity-check (LDPC) codes generally have good decoding performance in the waterfall region, but they exhibit higher error floors than regular ones. In this letter, we present a hybrid method, which combines code construction and the iterative decoding algorithm, to tackle this problem. Simulation results show that the proposed scheme decreases the error floor significantly for irregular LDPC codes over binary-input additive white Gaussian noise (BIAWGN) channel.

Adaptive network coded cooperation (ANCC) scheme may have excellent performance for data transmission from a large collection of terminals to a common destination in wireless networks. However, the random relay selection strategy for ANCC protocol may generate the distributed low-density parity-check (LDPC) codes with many short cycles which may cause error floor and performance degradation. In this paper, an optimized relay selection strategy for ANCC is proposed. Before data communication, by exploiting low-cost information interaction between the destination and terminals, the proposed method generates good assembles of distributed LDPC codes and its storage requirement reduces dramatically. Simulation results demonstrate that the proposed relay selection protocol significantly outperforms the random relay selection strategy.

In this letter, we propose a multi-stage decoding scheme with post-processing for low-density parity-check (LDPC) codes, which remedies the rapid performance degradation in the high signal-to-noise ratio (SNR) range known as error floor. In the proposed scheme, the unsuccessfully decoded words of the previous decoding stage are re-decoded by manipulating the received log-likelihood ratios (LLRs) of the properly selected variable nodes. Two effective criteria for selecting the probably erroneous variable nodes are also presented. Numerical results show that the proposed scheme can correct most of the unsuccessfully decoded words of the first stage having oscillatory behavior, which are regarded as a main cause of the error floor.

Trapping sets have been identified as one of the main factors causing error floors of low-density parity-check (LDPC) codes at high SNR values. By adding several new rows to the original parity-check matrix, a novel method is proposed to eliminate small trapping sets in the LDPC code's Tanner graph. Based on this parity-check matrix extension, we design new codes with low error floors from the original irregular LDPC codes. Simulation results show that the proposed method can lower the error floors of irregular LDPC codes significantly at high SNR values over AWGN channels.

In this letter, we propose a new modification to the belief propagation (BP) decoding algorithm for Finite-Geometry low-density parity-check (LDPC) codes. The modification is based on introducing feedback into the iterative process, which can break the oscillations of bit log-likelihood ratio (LLR) values. Simulations show that, with a given maximum iteration, the "feedback BP" (FBP) algorithm can achieve better performance than the conventional belief propagation algorithm.