Regular quasi-cyclic low density parity check codes with girth 8 from elementary number theory

This paper is concerned with (3, n) and (4, n) regular quasi-cyclic Low Density Parity Check (LDPC) code constructions from elementary number theory. Given the column weight, we determine the shift values of the circulant permutation matrices via arithmetic analysis. The proposed con-structions of quasi-cyclic LDPC codes achieve the following main advantages simultaneously: 1) our methods are constructive in the sense that we a-void any searching process; 2) our methods ensure no four or six cycles in the bipartite graphs corresponding to the LDPC codes; 3) our methods are direct constructions of quasi-cyclic LDPC codes which do not use any other quasi-cyclic LDPC codes of small length like component codes or any other algorithms/cyclic codes like building block; 4) the computations of the parameters involved are based on elementary number theory, thus very simple and fast. Simulation results show that the constructed regular codes of high rates perform almost 1.25 dB above Shannon limit and have no error floor down to the bit-error rate of 10 ^-6.