Structured quasi-cyclic LDPC codes with girth 18 and column-weight [formula omitted

Abstract

A class of maximum-girth geometrically structured quasi-cyclic (QC) low-density parity-check (LDPC) codes with column-weight 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 = { B 1 , B 2 , … , B l } is a configuration where B i is the subset of P specifying the set of nonzero block positions of the i th block-column of H . Let S = ( s i , j ) v × v be a matrix over Z m . 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 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 ⩽ 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 1.5 dB of Eb / N 0 .