130x1g2 - CCSDS

TM SYNCHRONIZATION AND CHANNEL CODING—SUMMARY OF CONCEPT AND RATIONALEFigure 8-5: Parity Check Matrix for the (n=1280, k=1024) AR4JA CodeThe AR4JA family of codes are built from a protograph shown in figure 8-4, with n {0,1,3}, where solid circles denote transmitted variable nodes, the open circle denotes apunctured variable node, and the circled crosses denote check nodes. These protographs areexpanded with circulants, in two stages, to build the parity check matrices. The firstcirculant expansion by a factor of 4 eliminates the parallel edges in the protograph, and thesecond expansion by the appropriate power of two creates the final parity check matrices.The circulants were chosen by randomized computer search and heuristics (reference [48]) toreduce the number of ‘trapping sets’; all parity check matrices are full rank. For example,the parity check matrix for the (n=1280, k=1024) rate 4/5 code is shown in figure 8-5. Thefinal 128 columns correspond to punctured symbols, and this puncturing increases the rate ofthe code from 8/11 to 4/5. Generator matrices for the AR4JA family, systematic in the first kpositions, can be computed using the usual matrix inverse method, though it iscomputationally burdensome for the larger cases.8.4 LDPC ENCODERSThe recommended LDPC codes are systematic in their first k symbols, and the following n-kparity symbols may be computed by multiplication by the parity portions of the densegenerator matrices. The number of binary operations required is proportional to k(n-k), butbecause the generator matrices have a block-circulant structure, their descriptionalcomplexity is proportional to n-k.CCSDS 130.1-G-2 Page 8-6 November 2012