Skriptum Kapitel 4

4 Anwendungen der Binären Blockcodes4.6.2 Beispiele für häufig verwendete (STANDARDISIERTE) CODES und deren GENERATORPOLYNOMEName GENERATOR POLYNOMIAL Periode | max. CW-Länge HA CW mit min. weightCRC-4 g 4 (x) = x 4 +x 3 +x 2 +x+1 (2 4 -1)/3 = 5 5 g 4 (x)CRC-7 g 7 (x) = x 7 +x 6 +x 4 +1 = (x 4 +x 3 +1)⋅(x 2 +x+1)⋅(x+1) k.g.V.(2 4 -1, 3, 1) = 15 4 g 7 (x)CRC-8 g 8 (x) = x 8 +x 7 +x 6 +x 4 +x 2 +1 = (x 5 +x 4 +x 3 +x 2 +1)⋅(x 2 +x+1)⋅(x+1) k.g.V.(2 5 -1, 3, 1) = 93 4 (x 2 +1)⋅g 8 (x)CRC-8-ATM g HEC (x) = x 8 +x 2 +x+1 = (x 7 +x 6 +x 5 +x 4 +x 3 +x 2 +1)⋅(x+1) k.g.V.(2 7 -1, 1) = 127 4 g HEC (x)CRC-10-ATM g OAM (x) = x 10 +x 9 +x 5 +x 4 +x+1 = (x 9 +x 4 +1)⋅(x+1) k.g.V.(2 9 -1, 1) = 511 4 (x 4 +1)/(x+1)⋅g OAM (x)CRC-12 g 12 (x) = x 12 +x 11 +x 3 +x 2 +x+1 = (x 11 +x 2 +1)⋅(x+1) k.g.V.(2 11 -1, 1) = 2 11 -1 = 2047 4 (x+1)⋅g 12 (x)CRC-ANSI g ANSI (x) = x 16 +x 15 +x 2 +1 = (x 15 +x+1)⋅(x+1) k.g.V.(2 15 -1, 1) = 2 15 -1 = 32767 4 g ANSI (x)CRC-CCITT 1) gCCITT(x) = x 16 +x 12 +x 5 +1 = (x 15 +x 14 +x 13 +x 12 +x 4 +x 3 +x 2 +x+1)⋅(x+1) k.g.V.(2 15 -1, 1) = 2 15 -1 = 32767 4 g CCITT (x)CRC-SDLC g SDLC (x) = x 16 +x 15 +x 13 +x 7 +x 4 +x 2 +x+1== (x 14 +x 13 +x 12 +x 10 +x 8 +x 6 +x 5 +x 4 +x 3 +x+1)⋅(x+1) 2 k.g.V.(2 14 -1, 2) = (2 14 -1)⋅2 = 32766 4 (x 16383 +1)⋅(x+1)CRC-24 g 24 (x) = x 24 +x 23 +x 14 +x 12 +x 8 +1 == (x 10 +x 8 +x 7 +x 6 +x 5 +x 4 +x 3 +x+1)⋅(x 10 +x 9 +x 6 +x 4 +1)⋅⋅(x 3 +x 2 +1)⋅(x+1) k.g.V.(2 10 -1, 7, 1) = (2 10 -1)⋅7 = 7161 6 g 24 (x)CRC-32A g 32A (x) = x 32 +x 30 +x 22 +x 15 +x 12 +x 11 +x 7 +x 6 +x 5 +x == (x 10 +x 9 +x 8 +x 6 +x 2 +x+1)⋅(x 10 +x 7 +x 6 +x 3 +1)⋅⋅(x 10 +x 8 +x 5 +x 4 +1)⋅(x+1)⋅x k.g.V.(2 10 -1, 2 10 -1, 2 10 -1, 1) = 1023 8 (?)⋅g 32A (x)CRC-32B g 32B (x) = x 32 +x 26 +x 23 +x 22 +x 16 +x 12 +x 11 +x 10 +x 8 +x 7 +x 5 +x 4 +x 2 +x+1 2 32 -1 ≈ 5⋅10 9 3 x 158594881 +x 58594881 modg 32B (x)IEC | TC57.1 g TC57.1 (x) = x 16 +x 14 +x 12 +x 11 +x 9 +x 8 +x 7 +x 4 +x+1k.g.V.(2 7 -1, 2 7 -1, 2) = 254= (x 7 +x 6 +x 3 +x+1)⋅(x 7 +x 6 +x 5 +x 4 +x 3 +x 2 +1)⋅(x+1) 2 für Blocklängen N ≤ 127IEC | TC57.2(nach Massey)46(x 127 +1)⋅(x+1)x 25 +x 25 modg(x)g TC57.2 (x) = x 16 +x 13 +x 12 +x 11 +x 10 +x 8 +x 6 +x 5 +x 2 +1 == (x 15 +x 14 +x 13 +x 11 +x 9 +x 8 +x 5 +x+1)⋅(x+1) k.g.V.((2 15 -1)/217, 1) = 151 6 x 38 +x 38 modg(x)1) Nach CCITT-Standard ist das CRC-Register am Beginn mit dem ALL_1-Wort, also mit FFh zu initialisieren. Letztlich ergibt sich dadurch ein COSET-Code.11.01.02 14:53 11