Parity bit

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Check Bit, which is a bit that can detect errors.

How

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• Examing the Least significant bit of the whole communication string
• Applied in an 8-bit communication system 7 + 1(parity)

the sum of every bit should be 0

Examples

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take 25 (0b11001) as an example

0	1	2	3	4	5	6	7
0	0	1	1	0	0	1	1

The parity bit is the 7^th bit, since the sum of every bit in 25 (0b11001) is 1; therefore the value of parity bit is $0 - 1 = 0 \oplus 1 = 1$

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take 30 (0b11110) as an example

0	1	2	3	4	5	6	7
0	0	1	1	1	1	0	0

The parity bit is the 7^th bit, since the sum of every bit in 30 (0b11110) is 0; therefore the value of parity bit is $0 - 0 = 0 \oplus 0 = 0$

Error Detection

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If single bit flip occurs, the sum of the bits will not be 0. That is, you can simply sum every bits to check whether an error appears.

a simple python code to check 30 (0b11110) is listed below

>>> from functools import reduce
>>> message = "00111100" # e = 0b0
>>> reduce(lambda a, b: int(a) ^ int(b), message) == 0
True
>>> message = "00110100" # e = 0b1000
>>> reduce(lambda a, b: int(a) ^ int(b), message) == 0
False

Problems

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The bigest problem of parity bit is that you can't identify the error if the hamming distance $H_d(m)$ is even. (We will say that this erorr detection only works while $H_d(m) \le 1$, even though it can detect $H_d(m) = 3$ since we are looking for the shortest).

Check m = 30 (0b11110) again but with $H_d(m) = 2$ with error e 11000000 => m + e = (0b1111110)

>>> from functools import reduce
>>> message = "11111100" # Message recieved in Binary
>>> reduce(lambda a, b: int(a) ^ int(b), message) == 0
True

• The result is True which means no errors however, $e \ne 0$

This is called a collision, which means two different system have the same parity code

See also

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• Numbers
• Summations
• Hamming Distance