Structured quasi-cyclic LDPC codes with girth 18 and column-weight J⩾3J⩾3

A class of maximum-girth geometrically structured quasi-cyclic (QC) low-density parity-check (LDPC) codes with column-weight J⩾3J⩾3 is presented. The method is based on the slope concept between two circulant permutation matrices and the concept of slope matrices.A LDPC code presented by a mv×mlmv×ml parity-check matrix HH, consisting of m×mm×m matrices each of which is either a circulant permutation matrix or a matrix with no nonzero entry, is called a mm-circulant vm×lmvm×lm LDPC code, or just a mm-circulant LDPC code. Let DD be a (v,J)(v,J) configuration; that is it has vv points, its blocks are of size JJ, and any two points are contained by at most one block. A mm-circulant LDPC code with a mv×mlmv×ml parity-check matrix HH is called a configuration-based   code if the set P={1,2,…,v}P={1,2,…,v} together with B={B1,B2,…,Bl}B={B1,B2,…,Bl} is a configuration where BiBi is the subset of PP specifying the set of nonzero block positions of the iith block-column of HH.Let S=(si,j)v×vS=(si,j)v×v be a matrix over ZmZm. Under a certain condition, the matrix SS is called a mm-slope-matrix   (mm-SM) over a given (v,J)(v,J) configuration DD. To any mm-SM SS over a (v,J)(v,J) configuration DD, with ll blocks, a DD-based mm-circulant vm×lmvm×lm LDPC code, referred to as a slope matrix (SM) code, is associated.It is shown that the maximum girth achieved by SM codes over a large class of configurations, including any balanced incomplete block design, is 18. A low-complexity algorithm producing such LDPC codes with girth 6⩽g⩽186⩽g⩽18 is given. As a few examples, a set of SM codes based on the Steiner triple systems STS(9) and STS(13), the 15-points 3×53×5 integer lattice, denoted L(3×5)L(3×5), and a 12-points configuration, denoted Aff*(16)Aff*(16), obtained from the 16-points affine plane Aff(16) are constructed. These codes have rates at least 0.25, 0.5, 0.4 and 0.37, respectively. From performance perspective, the constructed codes with girth g⩾14g⩾14 and length from 34,000 to 92,000 bits and the mentioned rates outperform the random-like LDPC codes of the same lengths and rates, and have a waterfall at about 10-610-6 BER and 1.5 dB of Eb/N0Eb/N0.