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

A class of maximum-girth geometrically structured quasi-cyclic (QC) low-density parity-check (LDPC) codes with columnweightJ3 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×ml parity-check matrix H, consisting of m×m matrices each of which is either a circulant permutation matrix or a matrix with no nonzero entry, is called am-circulant vm×lm LDPC code, or just am-circulant LDPC code. Let D be a(v,J) configuration; that is it has v points, its blocks are of size J, and any two points are contained by at most one block. Am-circulant LDPC code with a mv×ml parity-check matrix H is called a configuration-based code if the set P={1,2,...,v}together with B={B1, B2, ...,Bl}is a configuration where Bi is the subset of P specifying the set of nonzero block positions of the ith block-column of H. LetS=(si,j) v×v be a matrix over Zm. Under a certain condition, the matrix Sis called am-slope-matrix(m-SM) over a given (v,J) configurationD.Toanym-SM Sover a (v,J) configurationD, with l blocks, aD-basedm-circulantvm×lm LDPC code, referred to as aslope 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 6g18 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 3×5 integer lattice, denotedL(3×5), and a 12-points configuration, denotedAff ∗ (16), obtained from the 16-points affine planeAff(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 girthg14 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 −6 BER and 1.5 dB ofEb/N0. 2009 Elsevier GmbH. All rights reserved.

Quasi-cyclic LDPC codes; Circulant matrices; Block structure graph; Slope matrix codes; Configuration-based LDPC codes