Distribution Of Non-zero Elements Research Articles

FER Performance Analysis and Optimization of Diagonal Structure Based QC-LDPC Codes with Girth 12 Using LU Decomposition

The manuscript presents the construction of a diagonally structured Parity check matrix using lower and upper (LU) decomposition for regular Quasi Cyclic-Low Density Parity Check Codes (QC-LDPC) with girth 12. This diagonally structured parity check matrix reduces the sparsity characteristics of QC-LDPC codes. The regular Quasi Cyclic-LDPC code has been constructed using circulant shifting and randomization method in which uniform distribution of non-zero elements, XOR operation, and pairing of row and column has been done. A Simplified Log Domain Sum-Product Algorithm (SLDSPA) is used for the decoding of the constructed code. To reduce the CPU run time of the constructed code, a column-wise loop optimization technique is applied. The frame error rate and CPU run time is calculated for different code lengths such as code1 (816, 408), code2 (1008, 504), code3 (1032, 516), code4 (1944, 972), code5 (672, 336) and code6 (504, 252). The construction of regular QC-LDPC codes on half code rate is obtained for the weight of each row is 6 and each column is given by 3. The results showing that the constructed code is better performing in terms of Low FER (10−1 to 10−7), Low SNR (0 to 3.5 dB) and Less CPU run time in milliseconds.

MMSparse: 2D partitioning of sparse matrix based on mathematical morphology

Sparse matrix is any matrix with enough zeros that it pays to take advantage of them. The computational efficiency of sparse matrix–vector multiplication (SpMV) is significantly influenced by the distribution of non-zero elements in sparse matrix, which is not fully exploited by traditional one-dimensional and two-dimensional partitioning approaches. In this paper, we present a novel two-dimensional partitioning method based on mathematical morphology theory. The matrix is partitioned into dense and sparse areas utilizing some basic morphological transformations, including dilatation, filling, opening, and skeletonization. These dense areas are classified as rectangle, triangle, and diagonal according to their morphological features. We also propose a new MMSparse (Mathematical Morphological Sparse) format that stores each type of shapes in the most efficient format instead of one format for an entire matrix. We select different types of matrices from the SuiteSparse Matrix Collection and conduct a series of experiments on NVIDIA GTX 1080Ti GPU. The experimental results show that our approach achieves average speedup 2.47x over the best performance of the cuSPARSE library kernel, 2.54x over BCSR, 1.48x over yaSpMV, 1.85x over CSR5, 1.35x over the one-dimensional partition.

Towards optimal edge weight distribution and construction of field‐compatible low‐density parity‐check codes over GF( q )

Non-binary low-density parity-check (NB-LDPC) codes can be directly constructed by using algebraic methods, or indirectly constructed by mapping well-designed binary parity-check matrices to non-binary parity-check matrices. Given the Tanner graph (TG) of a NB-LDPC code, the selection of edge weights in the TG significantly affects the performance of the NB-LDPC code. The authors introduce an edge weight distribution (EWD) parameter for the TG of NB-LDPC codes. By utilising particle swarm optimisation (PSO), the EWD is optimised and it has been demonstrated that the optimal EWD approaches a two-element distribution for large field size and high average variable-node degree. With the optimised EWD, the authors construct a class of field-compatible LDPC (FC-LDPC) codes over GF(q) whose parity-check matrices only include elements 0, 1 and 2, and can be encoded and decoded over different field sizes. The simulations demonstrate that the performance of the proposed FC-LDPC codes improves monotonically with increasing field size, and significantly outperforms that of the corresponding algebraic NB-LDPC codes or NB-LDPC codes generated with uniform distribution of non-zero elements over GF(q).

Density of states of sparse random matrices

A supersymmetric formalism is used to derive a set of equations giving the density of states of any real, symmetric, sparse random matrix as a function of the distribution of non-zero elements and the mean number of non-zero elements per row, p. In the matrix where the non-zero elements take the values +or-1 with equal probability the equations are solved as p to infinity recovering results obtained previously with the replica method. As E to 0 the density of states rho (E) behaves as 1/E(ln(E))2. The more general case where +or-1 occur with unequal probabilities is also considered.