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Friday, 27 February 2009 By: Steve Warner

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Hamming Codes


Hamming Code (8:4)

The Hamming code used in teletext is used to reduce the possibility of errors for the sending address and control information.
It is possible to correct a single error in the received byte and detect multiple errors, when there are 2, 4 or 6 error bits. When there are 3, 5, 7 or 8 errors in a byte, this results in a false message being decoded.

The hamming code used in teletext is a (8,4) (n,k)

Code Rate Efficiency = k/n = 4/8 = 50%

Where D4 D3 D2 D1 are the data bits and C3 C2 C1 and C4 is the parity bit.

8 7 6 5 4 3 2 1
D4 C4 D3 C3 D2 C2 D1 C1



C1 = D1 Exclusive-OR D2 Exclusive-OR D4
C2 = D1 Exclusive-OR D3 Exclusive-OR D4
C3 = D2 Exclusive-OR D3 Exclusive-OR D4

Where Exclusive-OR represents an Exclusive-OR operation (addition without carry)


b8 b7 b6 b5 b4 b3 b2 b1
0 0 0 1 0 1 0 1
0 0 0 0 0 0 1 0
0 1 0 0 1 0 0 1
0 1 0 1 1 1 1 0
0 1 1 0 0 1 0 0
0 1 1 1 0 0 1 1
0 0 1 1 1 0 0 0
0 0 1 0 1 1 1 1
1 1 0 1 0 0 0 0
1 1 0 0 0 1 1 1
1 0 0 0 1 1 0 0
1 0 0 1 1 0 1 1
1 0 1 0 0 0 0 1
1 0 1 1 0 1 1 0
1 1 1 1 1 1 0 1
1 1 1 0 1 0 1 0









Message Bits
Hamming Code Protection Bits



b8 b7 b6 b5 b4 b3 b2 b1
A o o o o
B o o o o
C o o o o
D o o o o o o o o



Results of Parity Tests Inference Action
A,B,C D
All Correct Correct no errors accept message bits
All Correct Not Correct error in b7 accept message bits
Not all Correct Correct multiple errors reject message bits
Not all Correct Not Correct single errors refer to table above to identify error. Correct error if in message bit






Hamming Code (24:18)

 

Byte 1

C1 C2 D1 C3 D2 D3 D4 C4

Byte 2

D5 D6 C5 C6 C7 C8 C9 D7

Byte 3

D8 D9 D10 D11 D12 D13 D14 C10

 

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