High girth LDPC codes construction based on combinatorial design

A combinatorial-based girth-10 LDPC code design is presented. We use a 3D cyclic rectangular lattice to generate BIBDs (balanced incomplete block designs) by constructing planes and lines in the lattice. Using this structure, we can get a family of regular (n, 3, k) LDPC codes with girth of at least 10 explicitly. The main advantage is that the construction is well-structured, explicit and can lead to a very low complexity of implementation but rather good performance. We can get codes of a large range of lengths and rates by choosing a series of parameters of the lattice and give the best tradeoff between code rate and performance. Simulation results also show that the codes provide extremely high performance (about 0.45 dB to Shannon limit) over a binary-input additive white Gaussian noise (AWGN) channel.