A method is presented for constructing LDPC codes with excellent performance, simple hardware implementation, low encoder complexity, and which can be concisely documented. The simple code structure is achieved by using a base graph, expanded with circulants. The base graph is chosen by computer search using simulated annealing, driven by density evolution's decoding threshold as determined by the reciprocal channel approximation. To build a full parity check matrix, each edge of the base graph is replaced by a circulant permutation, chosen to maximize loop length by using a Viterbi-like algorithm.
A method is presented for constructing LDPC codes with excellent performance, simple hardware implementation, low encoder complexity, and which can be concisely documented. The simple code structure is achieved by using a base graph, expanded with circulants. The base graph is chosen by computer search using simulated annealing, driven by density evolution's decoding threshold as determined by the reciprocal channel approximation. To build a full parity check matrix, each edge of the base graph is replaced by a circulant permutation, chosen to maximize loop length by using a Viterbi-like algorithm.
Channel coding is a key component of artificial communication systems, allowing reliable communication using unreliable channels. In the last decade, iteratively decoded channel codes have become or clearly will become standards in a wide range of applications where large amounts of information must be communicated using unreliable media. Of the class of iteratively decoded codes, Low-density parity check (LDPC) codes are arguably the simplest class to describe, and indeed were described more than four decades ago in 1963 by Robert Gallager. The current understanding of LDPC codes has progressed in several significant ways beyond what had been expressed by 1963 by Gallager. Importantly, irregular LDPC codes, whose parity check matrices do not have constant row and column sums, have been shown to significantly outperform their regular counterparts explicitly considered by Gallager. By 1999, researchers had defined a class of irregular ensembles, each characterized by a pair of polynomials. Along with this new class of ensemble, they defined an analytical technique, density evolution, that accurately predicted the channel coding performance of a typical code under iterative message-passing decoding. The pair of polynomials could be effectively be designed by optimizing the coefficients of the polynomial for density evolution threshold threshold. This thesis concerns a different class of ensembles, namely protograph ensembles. Protograph ensembles are characterized by a template graph called, intuitively, a protograph. The Tanner-graph representation of a code in the ensemble is a random lift of the protograph. Protograph-based codes have significant advantages over unstructured irregular codes in regard to implementation of their encoders and decoders. In the decoder, this structure can be used in at least two distinct ways to organize the computations defined by any message-passing algorithm. If, in addition to the protograph structure, circulant structure is imposed on each "section" of the matrix then a quasi-cyclic code results, bestowing even mores advantages, especially in the possible implementation of the encoder. A central difficulty in using protograph ensembles is finding a suitable protograph. Since graphs are discrete objects, there is no obvious correspondence to any optimization model using vectors of real numbers. Instead, the technique of simulated annealing has been applied with a remarkable degree of success. For example, on the AWGN channel, given a constraint on the node degrees, protograph ensembles can be found that achieve a threshold only half as far (measured in dB) from the Shannon limit as unstructured irregular ensembles. This simultaneously illustrates an inherent performance advantage of protograph codes over unstructured codes as well as the efficacy of simulated annealing as an optimization technique. A persistent problem which appears to be common in all codes optimized for density evolution threshold is that of error floors. On a superficial level, this is explained by the maxim that "There's no such thing as a free lunch." In some contexts, such as in codes designed for the erasure channel, the phenomenon can be explained on a much deeper level, though it is not clear why the phenomenon should persist so universally. Still, even without a detailed understanding the cause of this problem, there are techniques that can mitigate error floors. An important tool toward this end is weight enumerators, which are discussed in chapter 3. Codeword and stopping set enumerators can be efficiently computed if a certain (non-concave) function can be efficiently maximized. Protographs that are selected on the basis of their enumerators have shown some success in reducing error floors.
To reduce the cycles in the Tanner graph which may affect the code performance,a recursive method to construct low-density parity-check(LDPC) codes was proposed.The method used a parity-check matrix of a short LDPC code as its mother matrix upon which a long code was constructed with circulant permutation matrices.By choosing the circulant permutation matrix parameters properly,the number of cycles having a given length in the Tanner graph of the constructed long codes were equal to or less than that of the short code.Either regular or irregular LDPC codes were constructed.Simulation results show that the LDPC codes constructed by this method have very low error-floor and almost no performance degradation due to their special structure as compared with random LDPC codes.
Low-Density Parity-Check (LDPC) codes are among the most effective modern error-correcting codes due to their excellent correction performance and highly parallel decoding scheme. Moreover, the nonbinary extension of such codes further increases performance in the short-block regime. In this paper, we review the key elements for the construction of implementation-oriented binary and nonbinary codes. These Quasi-Cyclic LDPC (QC-LDPC) codes additionally feature efficient encoder and decoder implementation frameworks. We then present a versatile algorithm for the construction of both binary and nonbinary QC-LDPC codes that have low encoding complexity and an optimized corresponding graph structure. Our algorithm uses a progressive edge growth algorithm, modified for QC-LDPC graph construction, and then performs an iterative global search for optimized cyclic shift values within the QC-LDPC circulants. Strong error correction performance is achieved by minimizing the number of short cycles, and cycles with low external connectivity, within the code graph. We validate this approach via error rate simulations of a transmission system model featuring an LDPC coder-decoder, digital modulation, and additive white Gaussian noise channels. The obtained numerical results validate the effectiveness of the proposed construction algorithm, with a number of constructed codes exhibiting either similar or superior performance to industry standard binary codes and selected nonbinary codes from the literature.
This paper provides a construction method for low-rate low density parity check codes. Inspired by recently proposed accumulate-repeat-accumulate (ARA) codes, and hybrid concatenated codes, in this paper we extend the construction to low rates. Such codes can be viewed as hybrid concatenations of simple modules such as accumulators, repetition codes, differentiators, and punctured single parity check codes. These codes constitute a subclass of LDPC codes with very fast encoder structure. They also have a projected graph or protograph representation that allows for high-speed decoder implementation. Based on density evolution, we show through some examples that low iterative decoding thresholds close to the channel capacity limits can be achieved, as the block size goes to infinity. Iterative decoding simulation results for short blocks are provided for a few examples that show near-capacity performance and very low error floor
This paper have developed farther the method of constructing sparse parity-check matrix (H-matrix) with determinate systemic structure as well as designing practice recursive encoding algorithm of the low-density parity-check (LDPC) codes. It discovers that the LDPC codes based on the circulant-shifted permutation Q-matrix created by N-queens searching algorithm perform better than them based on permutation identity matrix when Hd-matrix in H =[ HpHd] was constructed by using circulant-shifted permutation Q-matrix and identity matrix, respectively. It presents a simple and efficient algorithm of constructing Hdmatrix without 4-cycle and 6-cycle. The presented LDPC encoders require O(M) operation (M is check-bit size). They are rate-compatible and length- compatible encoders that can cover a very wide range of rate from 0.1 to 0.9 and length including any positive integer from 20 to infinity except prime number and multiple prime numbers. The presented LDPC codes with moderate length and 1/2 rate can achieve reliable performance - 10-5~10-6bit error rate (BER) - on an additive white Gaussian noise (AWGN) channel with signal-to-noise ratio (SNR) Eb/Noabout 1.42 dB1.
In this paper, we propose a novel method for constructing low-density parity-check (LDPC) codes that relies on circulant permutation matrices as the main principal elements. A key feature of this construction is the intentional avoidance of short cycles, particularly those of length four, which are known to degrade decoding performance. By eliminating these cycles, the approach contributes to lowering both encoding and decoding complexity, making the codes more efficient to implement. To assess their effectiveness, the proposed LDPC codes are evaluated and compared with uncoded binary phase shift keying (BPSK) transmission over an additive white Gaussian noise (AWGN) channel. The simulation outcomes demonstrate that the structured codes deliver competitive results, achieving error-rate performance similar to, and in some instances exceeding, well-known designs such as progressive edge growth (PEG) and quasi-cyclic (QC) codes.