The [24,12,8] extended Golay code is defined by the following generator matrix [4], [5]:
| | 11 01 11 01 11 00 00 00 00 00 00 00 | |
| | 00 11 11 10 01 11 00 00 00 00 00 00 | |
| | 00 00 11 01 10 11 11 00 00 00 00 00 | |
| | 00 00 00 11 01 11 01 11 00 00 00 00 | |
| | |
| | 00 00 00 00 11 01 11 01 11 00 00 00 | |
| G |
| | 00 00 00 00 00 00 11 01 10 11 11 00 | |
| | 00 00 00 00 00 00 00 11 01 11 01 11 | |
| | |
| | 11 00 00 00 00 00 00 00 11 01 11 01 | |
| | 01 11 00 00 00 00 00 00 00 11 11 10 | |
| | 10 11 11 00 00 00 00 00 00 00 11 01 | |
| | 01 11 01 11 00 00 00 00 00 00 00 11 | |
Similar to the [8,4,4] extended Hamming code, realization 9, one can find a shift register that is defining this code:
where
| | ui + z2 + z4 | if i = 0 or i = 3 (mod 4) | |
| f1(ui, z1, z2, z3, z4) |
| ui + z2 + z3 + z4 | if i = 1 (mod 4) |
| | ui + z1 + z2 + z4 | if i = 2 (mod 4) |
| f2(ui, z1, z2, z3, z4) |
| ui + z1 + z3 + z4 | if i = 0 or i = 3 (mod 4) |
| | ui + z1 + z2 + z3 + z4 | if i = 1 or i = 2 (mod 4) |
Note that now the shift register functions are time-variant. This shift register realizes a tail-biting trellis that consists of the following four sections three times repeated:
The realization finally looks as follows:
