Bit-flipping (BF) decoding of low-density parity-check codes is of low complexity but gives inferior performance in general. To improve performance and provide new BF decoder options for complexity-performance tradeoffs, we propose new designs for the flipping function (FF), the flipped bit selection (FBS) rule and the checksum weight updating schedule. The new FF adjusts the checksum weights in every iteration while our FBS rules take more information into account. These two modifications represent efforts to track more closely the evolutions of both check and variable nodes' reliabilities. Two selective update schedules are proposed to offer more performance and complexity tradeoffs.
The combinations of the new FBS rule and known FFs result in new BF decoders with improved performance and a modest complexity increase. On the other hand, combining the new FF and FBS rule gives a new decoder with performance comparable to that of the normalized min-sum algorithm while if we use a much simpler FBS rule instead, the decoder suffers little performance loss with reduced complexity. We also present a simple decision-theoretical argument to justify the new checksum weight formula and a time-expanded factor graph model to explain the proposed selective weight-updating schedules.
Characterizing the decoding failure rate of iteratively decoded Low- and Moderate-Density Parity Check (LDPC/MDPC) codes is paramount to build cryptosystems based on them, able to achieve indistinguishability under adaptive chosen ciphertext attacks. In this paper, we provide a statistical worst-case analysis of our proposed iterative decoder obtained through a simple modification of the classic in-place bit-flipping decoder. This worst case analysis allows both to derive the worst-case behaviour of an LDPC/MDPC code picked among the family with the same length, rate and number of parity check…
Iterative decoders used for decoding low-density parity-check (LDPC) and moderate-density parity-check (MDPC) codes are not characterized by a deterministic decoding radius and their error rate performance is usually assessed through intensive Monte Carlo simulations. However, several applications, like code-based cryptography, need guaranteed low values of the error rate, which are infeasible to assess through simulations, thus requiring the development of theoretical models for the error rate of these codes under iterative decoding. Some models of this type already exist, but become computat…
This paper is concerned with linear superposition systems in which all components of the superimposed signal are coded with an identical binary low-density parity-check (LDPC) code.
Reduced-Complexity Column-Layered Decoding and Implementation for LDPC Codes
分层译码是低密度奇偶校验（LDPC）译码器实现中的一种有效方法，它能以较低的计算复杂度获得较高的译码吞吐量。这项工作，第一次，解决了低复杂度的列分层解码方案和多Gb/s应用的VLSI架构。首先，将最小和算法引入到列分层解码中。然后探索算法变换和明智的近似以最大限度地降低总体计算复杂度。与原有的列分层译码相比，新方法在保持分层译码快速收敛速度的同时，可将高速LDPC码校验节点处理的计算复杂度降低90%以上。此外，一个宽松的流水线计划，使非常高的时钟速度的VLSI实现。利用这些新技术，本文设计了一种高效的准循环LDPC码译码器结构，并采用0.13 μ m CMOS工艺实现。结果表明，对于（4096，3584）LDPC码，最多迭代10次，译码吞吐量接近4Gb/s。因此，这项工作促进了列分层解码的实际应用，特别是使其在高速，高速率LDPC解码器的实现非常有吸引力。
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…
In this paper, we propose a Gradient Descent Bit-Flipping (GDBF) decoding with momentum, which considers past updates to provide inertia to the decoding process. We show that GDBF or randomized GDBF decoders with momentum may closely approach the floating-point Belief-Propagation decoding performance, and even outperform it in the error-floor region, especially for graphs with high connectivity degree.
In this paper, we focus on decoding nonbinary low-density parity-check (LDPC) codes in Galois fields of characteristic two via the proximal alternating direction method of multipliers (proximal-ADMM). By exploiting Flanagan/Constant-Weighting embedding techniques and the decomposition technique based on three-variables parity-check equations, two efficient proximal-ADMM decoders for nonbinary LDPC codes are proposed. We show that both of them are theoretically guaranteed convergent to some stationary point of the decoding model and either of their computational complexities in each proximal-AD…
The power consumption of the integrated circuit is becoming a significant burden, particularly for large-scale signal processing tasks requiring high throughput. The decoding process of LDPC codes is such a heavy signal processing task that demands power efficiency and higher decoding throughput. A promising approach to reducing both power and latency of a decoding process is to use an analog circuit instead of a digital circuit. This paper investigates a continuous-time gradient flow-based approach for decoding LDPC codes, which employs a potential energy function similar to the objective fun…
We propose a novel optimization-based decoding algorithm for LDPC-coded massive MIMO channels. The proposed decoding algorithm is based on a proximal gradient method for solving an approximate maximum a posteriori (MAP) decoding problem. The key idea is the use of a code-constraint polynomial penalizing a vector far from a codeword as a regularizer in the approximate MAP objective function. The code proximal operator is naturally derived from code-constraint polynomials. The proposed algorithm, called proximal decoding, can be described by a simple recursion consisting of the gradient descent …
In this paper, we propose a non-uniform windowed decoder for multi-dimensional spatially-coupled LDPC (MD-SC-LDPC) codes over the binary erasure channel. An MD-SC-LDPC code is constructed by connecting together several SC-LDPC codes into one larger code that provides major benefits over a variety of channel models. In general, SC codes allow for low-latency windowed decoding. While a standard windowed decoder can be naively applied, such an approach does not fully utilize the unique structure of MD-SC-LDPC codes. In this paper, we propose and analyze a novel non-uniform decoder to provide more…
We consider automorphism ensemble decoding (AED) of quasi-cyclic (QC) low-density parity-check (LDPC) codes. Belief propagation (BP) decoding on the conventional factor graph is equivariant to the quasi-cyclic automorphisms and therefore prevents gains by AED. However, by applying small modifications to the parity-check matrix at the receiver side, we can break the symmetry without changing the code at the transmitter. This way, we can leverage a gain in error-correcting performance using an ensemble of identical BP decoders, without increasing the worst-case decoding latency. The proposed met…
We study spatially coupled LDPC codes that allow access to sub-blocks much smaller than the full code block. Sub-block access is realized by a semi-global decoder that decodes a chosen target sub-block by only accessing the target, plus a prescribed number of helper sub-blocks adjacent in the code chain. This paper develops a theoretical methodology for analyzing the semi-global decoding performance of spatially coupled LDPC codes constructed from protographs. The main result shows that semi-global decoding thresholds can be derived from certain thresholds we define for the single-sub-block gr…
A new type of spatially coupled low-density parity-check (SC-LDPC) codes motivated by practical storage applications is presented. SC-LDPCL codes (suffix 'L' stands for locality) can be decoded locally at the level of sub-blocks that are much smaller than the full code block, thus offering flexible access to the coded information alongside the strong reliability of the global full-block decoding. Toward that, we propose constructions of SC-LDPCL codes that allow controlling the trade-off between local and global correction performance. In addition to local and global decoding, the paper develo…
