For a given block length and given degree sequences of the underlying Tanner graph (TG), the ensemble of short low-density parity-check (LDPC) codes can have considerable variation in performance. We present an efficient heuristic method to find good LDPC codes based on what we define as the girth distribution of the TG. This method can be used effectively to design short codes for applications where delay and complexity are of major concern.
For a given block length and given degree sequences of the underlying Tanner graph (TG), the ensemble of short low-density parity-check (LDPC) codes can have considerable variation in performance. We present an efficient heuristic method to find good LDPC codes based on what we define as the girth distribution of the TG. This method can be used effectively to design short codes for applications where delay and complexity are of major concern.
Low Density Parity Check (LDPC) codes have capacity-approaching performance over several channels of interest. In practice, for good block-error rate performance, the girth of the Tanner graph of an LDPC code needs to be as high as possible. In theory, to show that block-error rate approaches zero for increasing block-lengths, the girth of the Tanner graph sequence needs to tend to infinity with block-length. To meet these requirements, we construct sequences of large-girth irregular LDPC codes for a given degree-distribution pair (DDP) by applying a general node splitting algorithm on large girth graphs. The obtained Tanner graph meets the required DDP up to a suitable approximation. By optimizing the node-splitting method and using suitable large-girth graphs, we show code constructions with smaller block length for the same girth, when compared to previous constructions. Similar gains in block length are observed in the construction of sequences of large-girth protograph LDPC codes. Simulations, over a binary erasure channel, confirm the gains in block-error rate obtained by the large girth construction.
Short cycles in the Tanner graph of a given LDPC code has negative effect on the performance of iterative decoding. To improve the performance, we presented an approach for constructing Quasi-Cyclic Low-Density Parity-Check (QC-LDPC) codes whose associated Tanner graph has large girth and a small amount of short cycles. Each column block of the parity check matrix was designed by adding circulant permutation sub-matrices one after another, provided the current parity check matrix has minimal number of short cycles. The condition for the proposed QC-LDPC codes to have girth at least 6 was also analyzed. Regular and irregular QC-LDPC codes with various lengths and rates can be constructed. Simulations show that the proposed QC-LDPC codes have less short cycles and better performance than existing progressive edge growth based QC-LDPC codes.
In this paper, we study the cycle structure of quasi-cyclic (QC) low-density parity-check (LDPC) codes with the goal of obtaining the shortest code with a given degree distribution and girth. We focus on QC-LDPC codes, whose Tanner graphs are cyclic liftings of fully connected base graphs of size 3 × n, n ≥ 4, and obtain minimal lifting degrees that result in girths 6 and 8. This is performed through an efficient exhaustive search, and as a result, we also find all the possible non-isomorphic codes with the same minimum block length, girth, and degree distribution. The exhaustive search, which is ordinarily a formidable task, is made possible by pruning the search space of many codes that are isomorphic to those previously examined in the search process. Many of the pruning techniques proposed in this paper are also applicable to QC-LDPC codes with base graphs other than the 3 × n fully connected ones discussed here, as well as to codes with a larger girth. To further demonstrate the effectiveness of the pruning techniques, we use them to search for QC-LDPC codes with girths 10 and 12, and find a number of such codes that have a shorter block length compared with the best known similar codes in the literature. In addition, motivated by the exhaustive search results, we tighten the lower bound on the block length of QC-LDPC codes of girth 6 constructed from fully connected 3 × n base graphs, and construct codes that achieve the lower bound for an arbitrary value of n ≥ 4.
We propose a general method for constructing Tanner graphs having a large girth by establishing edges or connections between symbol and check nodes in an edge-by-edge manner, called progres- sive edge-growth (PEG) algorithm. Lower bounds on the girth of PEG Tanner graphs and on the minimum distance of the resulting low-density parity-check (LDPC) codes are derived in terms of parameters of the graphs. Simple variations of the PEG algorithm can also be applied to generate linear-time encodeable LDPC codes. Regular and irregular LDPC codes using PEG Tanner graphs and allowing symbol nodes to take values over GF are investigated. Simulation results show that the PEG algorithm is a powerful algorithm to generate good short-block-length LDPC codes.
A low-density parity-check (LDPC) code can be described by a Tanner graph composed of symbol nodes and check nodes. Short cycles in the Tanner graph should be avoided because such cycles will degrade the performance of the decoder and hence increase the error rate. In most of the circumstances, cycles of length 4 can be totally eliminated in the Tanner graph when the LDPC code has been designed properly. Locally, for each variable node in the Tanner graph, the girth may take the value of 6 or 8, or even higher. In this study, the authors attempt to increase the local girths of a Tanner graph with an aim to improving the error performance of the code. The authors propose and analyse the ‘degree spectrum’ of the symbol nodes connecting to each of the check nodes. Based on the analysis, the authors can determine if the local girths can be further increased. The authors also propose simple ways to construct codes with higher local girths. Finally, the authors show the characteristics and the error performance of the codes constructed with the proposed methods.
For a given block length sequences of the underlying Tanner graph (TG), the short circles of low-density parity check (LDPC) codes can have considerable variation in performance. By analyzing the shapes of the cycles of TG in parity check matrix, this paper presents a method of counting the number of 4-cycles, 6-cycles, 8-cycles and 10-cycles. Taking out a certain number of rows for different cycles, counting the number of cycles contained in these rows, then adding up all the number of cycles that contained in all possible combinations of the rows in the matrix, and we can get the number of cycles in the matrix. This method can be used effectively to evaluate the performance of LDPC codes according to their short circles distributions. Applying this method, we counting the number of cycles in the random LDPC codes and the quasi-cyclic LDPC codes, the BER performance shows that the random LDPC codes outperform the quasi-cyclic LDPC codes although their girth performance is not as good as the quasi-cyclic LDPC codes.