High-Rate QC LDPC Codes of Short and Moderate Length with Good Girth Profile

Irregular QC LDPC codes with parity-check matrices having different degree distributions are studied. A new algorithm for finding regular and irregular QC LDPC codes with a good girth profile as well as a good sliding-window girth is presented. As examples, simulation results for QC LDPC codes with good girth profile, rate R=4/5, and lengths about 1000, 2000, and 4000, constructed from base matrices with proper degree distributions are given. Their simulated BER and FER performances for belief propagation decoding are compared with the best previously known irregular QC LDPC codes of the same rate and length. It is shown that the constructed codes outperform the best previously known codes of same rate and lengths

Design of Non-Binary Quasi-Cyclic LDPC Codes by ACE Optimization

An algorithm for constructing Tanner graphs of non-binary irregular quasi-cyclic LDPC codes is introduced. It employs a new method for selection of edge labels allowing control over the code's non-binary ACE spectrum and resulting in low error-floor. The efficiency of the algorithm is demonstrated by generating good codes of short to moderate length over small fields, outperforming codes generated by the known methods.Comment: Accepted to 2013 IEEE Information Theory Worksho

A Greedy Search for Improved QC LDPC Codes with Good Girth Profile and Degree Distribution

Search algorithms for regular and irregular quasi-cyclic LDPC block codes with both good girth profile and good degree distribution are presented. New QC LDPC block codes of various code rates are obtained and their bit error rate performance is compared with that of the corresponding LDPC block codes defined in the IEEE 802.16 WiMAX standard of the same block length and code rate

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

On generalized LDPC codes for ultra reliable communication

Ultra reliable low latency communication (URLLC) is an important feature in future mobile communication systems, as they will require high data rates, large system capacity and massive device connectivity [11]. To meet such stringent requirements, many error-correction codes (ECC)s are being investigated; turbo codes, low density parity check (LDPC) codes, polar codes and convolutional codes [70, 92, 38], among many others. In this work, we present generalized low density parity check (GLDPC) codes as a promising candidate for URLLC. Our proposal is based on a novel class of GLDPC code ensembles, for which new analysis tools are proposed. We analyze the trade-o_ between coding rate and asymptotic performance of a class of GLDPC codes constructed by including a certain fraction of generalized constraint (GC) nodes in the graph. To incorporate both bounded distance (BD) and maximum likelihood (ML) decoding at GC nodes into our analysis without resorting to multi-edge type of degree distribution (DD)s, we propose the probabilistic peeling decoding (P-PD) algorithm, which models the decoding step at every GC node as an instance of a Bernoulli random variable with a successful decoding probability that depends on both the GC block code as well as its decoding algorithm. The P-PD asymptotic performance over the BEC can be efficiently predicted using standard techniques for LDPC codes such as Density evolution (DE) or the differential equation method. We demonstrate that the simulated P-PD performance accurately predicts the actual performance of the GLPDC code under ML decoding at GC nodes. We illustrate our analysis for GLDPC code ensembles with regular and irregular DDs. This design methodology is applied to construct practical codes for URLLC. To this end, we incorporate to our analysis the use of quasi-cyclic (QC) structures, to mitigate the code error floor and facilitate the code very large scale integration (VLSI) implementation. Furthermore, for the additive white Gaussian noise (AWGN) channel, we analyze the complexity and performance of the message passing decoder with various update rules (including standard full-precision sum product and min-sum algorithms) and quantization schemes. The block error rate (BLER) performance of the proposed GLDPC codes, combined with a complementary outer code, is shown to outperform a variety of state-of-the-art codes, for URLLC, including LDPC codes, polar codes, turbo codes and convolutional codes, at similar complexity rates.Programa Oficial de Doctorado en Multimedia y ComunicacionesPresidente: Juan José Murillo Fuentes.- Secretario: Matilde Pilar Sánchez Fernández.- Vocal: Javier Valls Coquilla

Construction of LDPC Codes Using Randomly Permutated Copies of Parity Check Matrix

Low-density parity-check codes (LDPC) have been shown to have good error correcting performance, putting in mind the Shannon's limit approaching capability. This enables an efficient and reliable communication. However, the construction method of LDPC code can vary over a wide range of parameters such as rate, girth and length. There is a need to develop methods of constructing codes over a wide range of rates and lengths with good performance. This research studies the construction of LDPC codes in randomized and structured form. The contribution of this thesis is introducing a method called "Randomly permutated copies of parity check matrix" that uses a base parity check matrix designed by a random or structured construction method such as Gallager or QC-LDPC codes respectively to get codes with multiple lengths and same rate of the base matrix. This is done by using a seed matrix with row and column weights of one, distributed randomly and can be addressed by a number in the base matrix. This method reduces the memory space needed for storing large parity check matrices, and also reduces the probability of failing to construct a parity matrix by random approaches. Numerical results show that the proposed construction performs similarly to random codes with the same length and rate as in Gallager's and Mackay's codes. It also increases the girth average of the Tanner graph and reduces the number of 4 cycles in the resulted matrix if exists in a base graph

Rate-compatible LDPC Codes based on Primitive Polynomials and Golomb Rulers

We introduce and study a family of rate-compatible Low-Density Parity-Check (LDPC) codes characterized by very simple encoders. The design of these codes starts from simplex codes, which are defined by parity-check matrices having a straightforward form stemming from the coefficients of a primitive polynomial. For this reason, we call the new codes Primitive Rate-Compatible LDPC (PRC-LDPC) codes. By applying puncturing to these codes, we obtain a bit-level granularity of their code rates. We show that, in order to achieve good LDPC codes, the underlying polynomials, besides being primitive, must meet some more stringent conditions with respect to those of classical punctured simplex codes. We leverage non-modular Golomb rulers to take the new requirements into account. We characterize the minimum distance properties of PRC-LDPC codes, and study and discuss their encoding and decoding complexity. Finally, we assess their error rate performance under iterative decoding

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

Sparse graph-based coding schemes for continuous phase modulations

The use of the continuous phase modulation (CPM) is interesting when the channel represents a strong non-linearity and in the case of limited spectral support; particularly for the uplink, where the satellite holds an amplifier per carrier, and for downlinks where the terminal equipment works very close to the saturation region. Numerous studies have been conducted on this issue but the proposed solutions use iterative CPM demodulation/decoding concatenated with convolutional or block error correcting codes. The use of LDPC codes has not yet been introduced. Particularly, no works, to our knowledge, have been done on the optimization of sparse graph-based codes adapted for the context described here. In this study, we propose to perform the asymptotic analysis and the design of turbo-CPM systems based on the optimization of sparse graph-based codes. Moreover, an analysis on the corresponding receiver will be done