High-rate irregular Generalized LDPC (GLDPC) codes with low-complexity constituent codes are studied. The main area of focus is the optimization of the underlying irregular binary LDPC codes and their matching with constituent codes. Finite length random coding bound on the maximum-likelihood decoding error probability of a random ensemble of irregular GLDPC codes is derived and compared with a similar bound for non-binary (NB) LDPC codes over small alphabets.A new algorithm for assigning columns of the parity-check matrix of a constituent code to nonzero elements of a base matrix of the quasi-cyclic (QC)-GLDPC code is presented. Examples of the constructed QC-GLDPC codes are given. The simulation results for frame error rate (FER) performance of the belief-propagation (BP) decoding for the QC-GLDPC codes are presented and compared with the same performance of NB QC-LDPC codes over small alphabets and binary LDPC codes in the WiFi and 5G standards.Both GLDPC codes and NB LDPC codes with sufficiently low decoding complexity are shown to outperform their binary LDPC counterparts, while GLDPC codes appear to be more advantageous in terms of decoding complexity.
High-rate irregular Generalized LDPC (GLDPC) codes with low-complexity constituent codes are studied. The main area of focus is the optimization of the underlying irregular binary LDPC codes and their matching with constituent codes. Finite length random coding bound on the maximum-likelihood decoding error probability of a random ensemble of irregular GLDPC codes is derived and compared with a similar bound for non-binary (NB) LDPC codes over small alphabets.A new algorithm for assigning columns of the parity-check matrix of a constituent code to nonzero elements of a base matrix of the quasi-cyclic (QC)-GLDPC code is presented. Examples of the constructed QC-GLDPC codes are given. The simulation results for frame error rate (FER) performance of the belief-propagation (BP) decoding for the QC-GLDPC codes are presented and compared with the same performance of NB QC-LDPC codes over small alphabets and binary LDPC codes in the WiFi and 5G standards.Both GLDPC codes and NB LDPC codes with sufficiently low decoding complexity are shown to outperform their binary LDPC counterparts, while GLDPC codes appear to be more advantageous in terms of decoding complexity.
We propose a novel approach to optimization of irregular nonbinary (NB) quasi-cyclic (QC)-LDPC codes over small alphabets. In this approach, first, the base parity-check matrices are constructed by a simulated annealing method, and then these matrices are labeled by the field elements while maximizing the so- called generalized girth of the Tanner graph. In order to analyze the performance of the constructed irregular NB LDPC codes, a new ensemble of irregular NB LDPC codes over the extensions of the binary Galois field is introduced. A finite-length random coding bound on the error probability of the maximum-likelihood (ML) decoding over the binary phase shift keying (BPSK) input AWGN channel for the new code ensemble is derived. The frame error rate (FER) performance of the sum-product belief-propagation (BP) decoding of the constructed NB QC-LDPC block codes is compared to that of both the optimized binary QC-LDPC block codes in the 5G standard and the best known NB QC-LDPC codes as well as to the derived random coding bound on the ML decoding error probability. It is shown that the obtained bound predicts the behavior of BP decoding performance of practical NB QC-LDPC codes more accurately than the BP decoding thresholds do.
We propose a novel approach for optimization of nonbinary (NB) quasi-cyclic (QC)-LDPC codes. In this approach, the base parity-check matrices are constructed by the simulated annealing method, and then labeled while maximizing the so-called generalized girth of the NB LDPC code Tanner graph. Random coding bounds on the ML decoding error probability for ensembles of "almost regular" NB LDPC codes of finite lengths over extensions of the binary Galois field are derived. These bounds are based on the average bit weight spectra for the ensembles of NB LDPC codes. The observed FER performance of the sum-product BP decoding of "almost regular" NB QC-LDPC block codes is presented and compared to the finite-length random coding bounds, as well as to the performance of the optimized binary QC-LDPC block code in the 5G standard. In the waterfall region, the gap between the finite-length bounds on the error probability of the ML decoding and the simulation performance of the BP decoding is about 0.1 – 0.2 dB.
We propose an approach for optimizing nonbinary (NB) quasi-cyclic (QC) LDPC codes. This approach combines constructing of base parity-check matrices by simulated annealing and labeling the obtained base matrices aimed at maximizing the so-called generalized girth of the NB LDPC code Tanner graph. Tightened random coding bounds based on the average binary spectra for ensembles of "almost regular" NB LDPC codes of finite lengths over extensions of the binary Galois field are derived. The simulated FER performance of the sum-product BP decoding of "almost regular" NB QC-LDPC block codes are presented and compared with the derived finite-length random coding bounds as well as with the same performance of the optimized binary QC-LDPC block code in the 5G standard. In the waterfall region our finite-length bounds on the error probability of ML decoding are about 0.1--0.2 dB away from the simulated FER performance of BP decoding.
Generalized low-density parity-check (GLDPC) codes have emerged as a promising class of error-correcting codes that extend the capabilities of traditional LDPC codes by incorporating more powerful component codes. This paper investigates an approach to construct long GLDPC codes using short component codes. In this scheme, the BCJR algorithm is used as a component decoder and integrated into the belief propagation algorithm. Simulation results demonstrate that constructions incorporating quasi-cyclic LDPC codes yield the best performance. Compared to the conventional decoding approach for LDPC codes, the proposed generalized constructions demonstrated better bit and frame error rate performance. This approach is promising for communication systems that require low latency. A comparison is also provided with LDPC codes constructed using the Progressive Edge-Growth (PEG) algorithm.
Generalized low-density parity-check (GLDPC) codes, where single parity-check constraints on the code bits are replaced with generalized constraints (an arbitrary linear code), are a promising class of codes for low-latency communication. The block error rate performance of the GLDPC codes, combined with a complementary outer code, has been shown to outperform a variety of state-of-the-art code and decoder designs with suitable lengths and rates for the 5G ultra-reliable low-latency communication (URLLC) regime. A major drawback of these codes is that it is not known how to construct appropriate polynomial matrices to encode them efficiently. In this paper, we analyze practical constructions of quasi-cyclic GLDPC (QC-GLDPC) codes and show how to construct polynomial generator matrices in various forms using minors of the polynomial matrix. The approach can be applied to fully generalized matrices or partially generalized (with mixed constraint node types) to find better performance/rate trade-offs. The resulting encoding matrices are presented in useful forms that facilitate efficient implementation. The rich substructure displayed also provides us with new methods of determining low weight codewords, providing lower and upper bounds on the minimum distance and often giving those of weight equal to the minimum distance. Based on the minors of the polynomial parity-check matrix, we also give a formula for the rank of any parity-check matrix representing a QC-LDPC or QC-GLDPC code, and hence, the dimension of the code. Finally, we show that by applying double graph-liftings, the code parameters can be improved without affecting the ability to obtain a polynomial generator matrix.
There is an important breakthrough in the field of forward error correction techniques, low-density parity-check (LDPC) codes. These codes have potential to approach the Shannon limit with reliable performance on a given channel, known as the channel capacity. Recently, researchers pay much more attention towards LDPC codes because its performance is better than Turbo codes. However, the LDPC code is the best when it is an irregular code and uses with a very large block-length. Although, the short block-length LDPC codes perform not so well, they are easy to implement in various practical applications. Therefore, there is still a demand of new development on the encoding side of LDPC codes for various ranges of code rate and code length so as to capable of using in many applications.This thesis proposes the development of quasi-cyclic (QC) LDPC codes for column weight 3. In the first part of this thesis, we propose a new construction algorithm of QC-LDPC codes for medium to large block-length by combining QC-LDPC codes of small block-length as their component codes, via Chinese remainder theorem (CRT). Such component codes were constructed by permuting each column block sequentially to attain the desire local girth. After combining all component codes to generate an expanded parity-check matrix, the resulting girth is greater than or at least equal to the maximum girth of the component codes. Simulation results show that our proposed construction method of the parity-check matrix significantly outperforms the other well-known existing methods in terms of low error-floor, simple structure, high performance, and can reduce encoding complexity.In addition, this thesis also proposes two new construction methods for QC-LDPC codes, namely a base matrix method and a subtraction based method. A base matrix based method is a simple, less computational complexity method for constructing the exponent matrix (3, K) of girth 8, 10 and 12 of QC-LDPC codes. Another method is a subtraction method, which has a similar exponent matrix as a base matrix method and has a girth at least 8. Results indicate that the LDPC codes constructed from these two methods have flexibility for arbitrary block-column length K and have a similar BER performance if compared to existing methods.