An Efficient Algorithm for Counting Cycles in QC and APM LDPC Codes

In this paper, a new method is given for counting cycles in the Tanner graph of a (Type-I) quasi-cyclic (QC) low-density parity-check (LDPC) code which the complexity mainly is dependent on the base matrix, independent from the CPM-size of the constructed code. Interestingly, for large CPM-sizes, in comparison of the existing methods, this algorithm is the first approach which efficiently counts the cycles in the Tanner graphs of QC-LDPC codes. In fact, the algorithm recursively counts the cycles in the parity-check matrix column-by-column by finding all non-isomorph tailless backtrackless closed (TBC) walks in the base graph and enumerating theoretically their corresponding cycles in the same equivalent class. Moreover, this approach can be modified in few steps to find the cycle distributions of a class of LDPC codes based on Affine permutation matrices (APM-LDPC codes). Interestingly, unlike the existing methods which count the cycles up to $2g-2$, where $g$ is the girth, the proposed algorithm can be used to enumerate the cycles of arbitrary length in the Tanner graph. Moreover, the proposed cycle searching algorithm improves upon various previously known methods, in terms of computational complexity and memory requirements.Comment: 18 pages, 4 figure

4-Cycle Free Spatially Coupled LDPC Codes with an Explicit Construction

Spatially coupled low-density parity-check (SC-LDPC) codes are a class of capacity approaching LDPC codes with low message recovery latency when a sliding window decoding is used. In this paper, we first present a new method for the construction of a class of SC-LDPC codes by the incidence matrices of a given non-negative integer matrix $E$, and then the relationship of 4-cycles between matrix $E$ and the corresponding SC-LDPC code are investigated. Finally, by defining a new class of integer finite sequences, called {\it good sequences}, for the first time, we give an explicit method for the construction of a class of 4-cycle free SC-LDPC codes that can achieve (in most cases) the minimum coupling width

Spatially Coupled LDPC Codes Constructed from Protographs

In this paper, we construct protograph-based spatially coupled low-density parity-check (SC-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 (BEC) and the binary-input additive white Gaussian noise channel (AWGNC) saturate to a particular value significantly better than the BP decoding threshold and numerically indistinguishable from the optimal maximum a-posteriori (MAP) 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 (MBS) channels with low-complexity BP decoding.Comment: Submitted to the IEEE Transactions on Information Theor

Codes on Graphs and More

Modern communication systems strive to achieve reliable and efficient information transmission and storage with affordable complexity. Hence, efficient low-complexity channel codes providing low probabilities for erroneous receptions are needed. Interpreting codes as graphs and graphs as codes opens new perspectives for constructing such channel codes. Low-density parity-check (LDPC) codes are one of the most recent examples of codes defined on graphs, providing a better bit error probability than other block codes, given the same decoding complexity. After an introduction to coding theory, different graphical representations for channel codes are reviewed. Based on ideas from graph theory, new algorithms are introduced to iteratively search for LDPC block codes with large girth and to determine their minimum distance. In particular, new LDPC block codes of different rates and with girth up to 24 are presented. Woven convolutional codes are introduced as a generalization of graph-based codes and an asymptotic bound on their free distance, namely, the Costello lower bound, is proven. Moreover, promising examples of woven convolutional codes are given, including a rate 5/20 code with overall constraint length 67 and free distance 120. The remaining part of this dissertation focuses on basic properties of convolutional codes. First, a recurrent equation to determine a closed form expression of the exact decoding bit error probability for convolutional codes is presented. The obtained closed form expression is evaluated for various realizations of encoders, including rate 1/2 and 2/3 encoders, of as many as 16 states. Moreover, MacWilliams-type identities are revisited and a recursion for sequences of spectra of truncated as well as tailbitten convolutional codes and their duals is derived. Finally, the dissertation is concluded with exhaustive searches for convolutional codes of various rates with either optimum free distance or optimum distance profile, extending previously published results

ADVANCED SIGNAL PROCESSING FOR MAGNETIC RECORDING ON PERPENDICULARLY MAGNETIZED MEDIA

In magnetic recording channels (MRCs) the readback signal is corrupted by many kinds of impairments, such as electronic noise, media noise, intersymbol interference (ISI), inter-track interference (ITI) and different types of erasures. The growth in demand for the information storage, leads to the continuing pursuit of higher recording density, which enhances the impact of the noise contamination and makes the recovery of the user data from magnetic media more challenging. In this dissertation, we develop advanced signal processing techniques to mitigate these impairments in MRCs.We focus on magnetic recording on perpendicularly magnetized media, from the state-of-the art continuous media to bit-patterned media, which is a possible choice for the next generation of products. We propose novel techniques for soft-input soft-output channel detection, soft iterative decoding of low-density parity-check (LDPC) codes as well as LDPC code designs for MRCs.First we apply the optimal subblock-by-subblock detector (OBBD) to nonbinary LDPC coded perpendicular magnetic recording channels (PMRCs) and derive a symbol-based detector to do the turbo equalization exactly. Second, we propose improved belief-propagation (BP) decoders for both binary and nonbinary LDPC coded PMRCs, which provide significant gains over the standard BP decoder. Third, we introduce novel LDPC code design techniques to construct LDPC codes with fewer short cycles. Performance improvement is achieved by applying the new LDPC codes to PMRCs. Fourth, we do a substantial investigation on Reed-Solomon (RS) plus LDPC coded PMRCs. Finally, we continue our research on bit-patterned magnetic recording (BPMR) channels at extremely high recording densities. A multi-track detection technique is proposed to mitigate the severe ITI in BPMR channels. The multi-track detection with both joint-track and two-dimensional (2D) equalization provide significant performance improvement compared to conventional equalization and detection methods

Low-Density Parity-Check Coded High-order Modulation Schemes

In this thesis, we investigate how to support reliable data transmissions at high speeds in future communication systems, such as 5G/6G, WiFi, satellite, and optical communications. One of the most fundamental problems in these communication systems is how to reliably transmit information with a limited number of resources, such as power and spectral. To obtain high spectral efficiency, we use coded modulation (CM), such as bit-interleaved coded modulation (BICM) and delayed BICM (DBICM). To be specific, BICM is a pragmatic implementation of CM which has been largely adopted in both industry and academia. While BICM approaches CM capacity at high rates, the capacity gap between BICM and CM is still noticeable at lower code rates. To tackle this problem, DBICM, as a variation of BICM, introduces a delay module to create a dependency between multiple codewords, which enables us to exploit extrinsic information from the decoded delayed sub-blocks to improve the detection of the undelayed sub-blocks. Recent work shows that DBICM improves capacity over BICM. In addition, BICM and DBICM schemes protect each bit-channel differently, which is often referred to as the unequal error protection (UEP) property. Therefore, bit mapping designs are important for constructing pragmatic BICM and DBICM. To provide reliable communication, we have jointly designed bit mappings in DBICM and irregular low-density parity-check (LDPC) codes. For practical considerations, spatially coupled LDPC (SC-LDPC) codes have been considered as well. Specifically, we have investigated the joint design of the multi-chain SC-LDPC and the BICM bit mapper. In addition, the design of SC-LDPC codes with improved decoding threshold performance and reduced rate loss has been investigated in this thesis as well. The main body of this thesis consists of three parts. In the first part, considering 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. Furthermore, we jointly optimize degree distributions and channel assignments of LDPC codes using protograph-based extrinsic information transfer charts. In addition, we proposed a constrained progressive edge growth-like algorithm to jointly construct LDPC codes and bit mappings for DBICM, taking the capacity of each bit-channel into account. Simulation results demonstrate that the designed LDPC-coded DBICM systems significantly outperform LDPC-coded BICM systems. In the second part, we proposed a windowed decoding algorithm for DBICM, which uses the extrinsic information of both the decoded delayed and undelayed sub-blocks, to improve the detection for all sub-blocks. We show that the proposed windowed decoding significantly outperforms the original decoding, demonstrating the effectiveness of the proposed decoding algorithm. In the third part, we apply multi-chain SC-LDPC to BICM. We investigate various connections for multi-chain SC-LDPC codes and bit mapping designs and analyze the performance of the multi-chain SC-LDPC codes over the equivalent binary erasure channels via density evolution. Numerical results demonstrate the superiority of the proposed design over existing connected-chain ensembles and over single-chain ensembles with the existing bit mapping design

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum