Layered Construction Of QC-LDPC Codes On 3-Dimensional Rectangular Lattices

Quasi-Cyclic Low-Density Parity-Check(QC-LDPC)codes have attracted considerable attention of the researcher and are used in related standards because of their encoding simplicity as well as outstanding error correction performance.It is known that if there are small cycles in the Tanner graph of LDPC codes,it is not conducive to the convergence of the iterative algorithm.Therefore,the girth is an important factor when constructing the LDPC codes.At present,many construction methods consider the factor of girth.However,large girth does not guarantee the absence of small trapping sets,which can not guarantee a lower error floor.In this paper,the small trapping sets mainly include(a ? 8,b ? 3).For a given column weight,increasing the girth of a Tanner graph requires either increasing the number of variable nodes,thus requiring a longer code,or decreasing the row weight and increasing the number of check nodes,which lowers the code rate.Therefore,to preserve the length and rate while lowering the error floor,a code must be optimized not by simply increasing the girth but rather by more surgically avoiding the most harmful trapping sets.The construction method of regular column-weight-three QC-LDPC codes without small trapping sets is mainly studied,and a layered construction of LDPC codes on 3-dimensional rectangular lattices is explored in this paper.By analyzing the relationship between short cycles and(5,3),(6,4),(7,3)trapping sets of layered LDPC codes,and deriving the governing conditions of in-layer polygons and interlayer polygons,and elaborately selecting the slopes of in-layer and inter-layer lines,all the 4-cycles,6-cycles as well as inter-layer 8-cycles are removed,which results in a class of LDPC codes without(a ? 8,b ? 3)small trapping sets.In order to design a class of LDPC codes with girth 10,the Golomb ruler is adopted to further eliminate in-layer 8-cycles.A length-275(3,5)and a length-468(3,6)LDPC code with girth 8,as well as a length-575(3,5)and a length-1332(3,6)LDPC code with girth10 are constructed.By comparison the error rate performances of layered construction codes and random-lifting codes with the same girth on the additive Gaussian white noise channel decoded with the belief propagation algorithm,it can be concluded that the codes in this paper exhibit lower error floors than those of their counterparts.