130x1g2 - CCSDS

TM SYNCHRONIZATION AND CHANNEL CODING—SUMMARY OF CONCEPT AND RATIONALE32, thus defining a regular-(4,32) LDPC code. This parity check matrix has two linearlydependent rows, so it has rank 1020.A generator matrix for this code has size 7156 × 8176. When the first 7154 positions aremade systematic, then the first 7154 rows can be written in the block-circulant form shown infigure 6-2 of reference [3]. The remaining two rows can be chosen as,⎡g⎢⎣g71557156⎤ ⎡0⎥ = ⎢⎦ ⎣0511×14511×141051151101511511⎤⎥⎦where X n represents the digit X repeated n times. This construction is not sufficient to makethe dense circulants B i,j unique, because g 7155 and g 7156 may be added arbitrarily to each. Thecirculants were chosen so that their upper right corners are ‘0’, making each of thehexadecimal numbers in table A-1 of [3] even. The code is then expurgated by leavingdimensions g 7155 and g 7156 unused.Equivalently, one could expurgate the code defined by H by adding two linearly independentconstraints,⎡h⎢⎣h10231024⎤ ⎡0⎥ = ⎢⎦ ⎣0511511005115110051151100511511015115111151151110511511015115111051151111511511115115110151151101511511115115111051151101511511⎤⎥⎦The block-circulant generator matrix for this code, systematic in the first 7154 positions, isunique and matches that described above.This expurgated code is shortened by setting the first 18 information bits to ‘zero’ (and nottransmitting them), and extended by appending two ‘zeros’. The resulting code, C2, has size(n=8160, k=7136). These parameters are multiples of 32 for convenience with 32-bitmicroprocessors, and also match the size of the (255, 233) Reed-Solomon code with aninterleave depth of I=4.Figure 8-4: Protographs for the AR4JA Code FamilyCCSDS 130.1-G-2 Page 8-5 November 2012