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Detecting Error Using Hamming Code

When the 12-bits data are read from the memory then be evaluated by follows

C1 = XOR of bits (1,3,5,7,9, 11)
C2 = XOR of bits (2,3,6,7,10,11)
C4 = XOR of bits (4,5,6,7,12)
C8 = XOR of bits (8,9,10,11,12)

then C in arrangement C=C₈C₄C₂C₁ be evaluated, if C=0000, indicates that no error has occured.

For example,

Evaluating the XOR of the corresponding bits, we determine the 4 check bits to be as follows:
                            C₈  C₄  C₂  C₁
       For no errorr : 0     0    0    0
With error in bit 1 : 0     0    0    1
With error in bit 5 : 0     1    0    1

The Hamming code can detect and correct only a single error. By adding another parity bit to the coded word, the Hamming code can be used to correct a single error and detect double errors. If we include this additional parity bit, then the previous 12-bi t coded word becomes
0011 1001 0100P₁₃, where P₁₃ is evaluated from the exclusive-OR of the other 12 bits. This produces the 13-bit word 0011100101001(even parity). When the 13-bit word is read from memory, the check bits are evaluated, as is the parity P over the entire 13 bits. If P = 0, the parity is correct (even parity), but if P = 1, then the parity over the 13 bits is incorrect (odd
parity). The following four cases can arise:

If C = 0 and P = 0, no error occurred.
If C # 0 and P = 1, a single error occurred that can be corrected.
If C # 0 and P = 0, a double error occurred that is detected, but that cannot be corrected.
If C = 0 and P = 1, m error occurred in the P13 bit.

　Source: Digital Design, M.Morris Mano

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