Structuredquasi - cyclicLDPCcodeswith girth 18 and column - weight J 3

A class of maximum-girth geometrically structured quasi-cyclic (QC) low-density parity-check (LDPC) codes with columnweight J 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 ×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 a m-circulant vm × lm LDPC code, or just a m-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. A m-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 i th block-column of H . Let S = (si, j )v×v be a matrix over Zm . Under a certain condition, the matrix S is called a m-slope-matrix (m-SM) over a given (v, J ) configuration D. To any m-SM S over a (v, J ) configuration D, with l blocks, a D-based m-circulant vm × lm LDPC code, referred to as a slope matrix (SM) code, is associated. It is shown that themaximumgirth 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 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× 5 integer lattice, denoted L(3× 5), and a 12-points configuration, denoted 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 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 10−6 BER and 1.5dB of Eb/N0. 2009 Elsevier GmbH. All rights reserved.