# Synchronization and channel coding 734 encoding note

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Unformatted text preview: ure of the quasi-cyclic encoder and increase complexity of the generator matrix for the basic (8176,7156) LDPC code. They are not included in this specification, which results in a generator matrix for a systematic (8176,7154) subcode that can be constructed entirely of circulants as shown in figure 7-2. 7.3.4.1 The generator matrix for the systematic (8176,7154) subcode shall be that illustrated in figure 7-2. 7.3.4.2 The left portion of the matrix shall be a 7154 × 7154 identity matrix, shown here as a block matrix, where I denotes the identity matrix of size 511 × 511, and 0 denotes the allzero matrix of size 511 × 511. 7.3.4.3 The right portion of the matrix shall contain two columns of 511 × 511 circulants, denoted Bi,j, and constructed as follows: ⎡ A 1,15 1) D = ⎢ ⎣A 2,15 A 1,16 ⎤ shall be defined from figure 7-1 and table 7-1. A 2,16 ⎥ ⎦ NOTE – This equation describes a 1022 × 1022 matrix. 2) u = (1 0 0 0 …. 0) shall be the unit 511 tuple, i.e., a vector quantity of length 511 with a ‘1’ at the leftmost position and ‘0’s in the rest. 3) zi = (bi,1 bi,2) shall be defined, where i = 1, 2, ..., 14 and the bi,js are the first rows of the Bi,j circulants. NOTE – For i = 1, 2, ..., 14, zi is a vector with 1020 elements. ⎡A ⎤ 4) M i = ⎢ 1,i ⎥ shall be defined, where i = 1, 2, …, 14. ⎣A 2,i ⎦ NOTE – The parity check matrix can now be represented as: [M1 M2 … M14 D]. 5) The 511th and 1022nd elements of zi shall be set to zero and M i u T + Dz iT = 0 shall be solved for zi, where i = 1, 2, …, 14 and T superscript represents matrix transpose. NOTE – Since the rank of D is 1020 and not 1022, there are two linearly dependent columns. These columns can be taken to be the 511th and 1022nd. 6) The bi,js shall be extracted from the zis. NOTE – The bi,js are numerically tabulated in annex C. CCSDS 131.0-B-2 Page 7-5 August 2011 CCSDS RECOMMENDED STANDARD FOR TM SYNCHRONIZATION AND CHANNEL CODING ⎡I ⎢0 ⎢ ⎢0 ⎢ ⎢0 ⎢0 ⎢ ⎢0 ⎢0 ⎢ ⎢0 ⎢0 ⎢ ⎢0 ⎢ ⎢0 ⎢0 ⎢ ⎢0 ⎢0 ⎣ 0000000000000 I000000000000 0I00000000000 B1,1 B 2,1 B3,1 00I0000000000 000I000000000 B 4,1 B5,1 0000I00000000 00000I0000000 000000I000000 B 6,1 B 7,1 B8,1 0 0 0 0 0 0 0 I 0 0 0 0 0 B9,1 0 0 0 0 0 0 0 0 I 0 0 0 0 B10,1 0 0 0 0 0 0 0 0 0 I 0 0 0 B11,1 0 0 0 0 0 0 0 0 0 0 I 0 0 B12,1 0 0 0 0 0 0 0 0 0 0 0 I 0 B13,1 000000000000 I B14,1 B1,2 ⎤ B 2,2 ⎥ ⎥ B3,2 ⎥ ⎥ B 4,2 ⎥ B5,2 ⎥ ⎥ B 6,2 ⎥ B 7,2 ⎥ ⎥ B8,2...
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