Building on what you have learnt about parity bits we are now going to see a system that not only allows you to detect if the data you have been sent is incorrect, but it will allow you to correct the error. The way hamming code does this is to use multiple check digits in the same piece of sent data.
Highlight the column headings that are powers of 2 (1,2,4,8), these are the parity bits
Insert your data and highlight the parity bits
Work your way through the parity bits
2^0 = 1 : check 1, skip 1, check 1, skip 1 ... write down whether it's odd or even parity
2^1 = 2 : check 2, skip 2, check 2, skip 2 ... write down whether it's odd or even parity
2^2 = 4 : check 4, skip 4, check 4, skip 4 ... write down whether it's odd or even parity
etc..
Example: Odd Parity Hamming Code Check
Let's take a look a an example of data sent with odd parity
11
10
09
08
07
06
05
04
03
02
01
number the columns and highlight the powers of 2
1
0
0
0
0
0
0
1
1
1
1
insert your data
1
0
0
0
0
0
0
1
1
1
1
highlight the check bits
1
0
0
0
1
1
taking the 1st power of 2^0 (1) check 1 skip 1 = odd parity
1
0
0
0
1
1
taking the 2nd power of 2^1 (2) check 2 skip 2 = odd parity
0
0
0
1
taking the 3rd power of 2^2 (4) check 4 skip 4 = odd parity
1
0
0
0
taking the 4th power of 2^3 (8) check 8 skip 8 = odd parity
Note that for the check 8 skip 8 we ran out of digits, not to worry, take it as far as the bits given allow. As we can see each line is odd parity, and the sent data was supposed to be odd parity, this number is correct.
Exercise: Even Parity Hamming Code Question
Now try this example with even parity:
10101100011
Answer:
11
10
09
08
07
06
05
04
03
02
01
number the columns and highlight the powers of 2
1
0
1
0
1
1
0
0
0
1
1
insert your data
1
0
1
0
1
1
0
0
0
1
1
highlight the check bits
1
1
1
0
0
1
taking the 1st power of 2^0 (1) check 1 skip 1 = even parity
1
0
1
1
0
1
taking the 2nd power of 2^1 (2) check 2 skip 2 = even parity
1
1
0
0
taking the 3rd power of 2^2 (4) check 4 skip 4 = even parity
1
0
1
0
taking the 4th power of 2^3 (8) check 8 skip 8 = even parity
All are even parity, the data should be even parity, therefore it has been sent and received correctly
11011110010 being sent with odd parity
Answer:
11
10
09
08
07
06
05
04
03
02
01
number the columns and highlight the powers of 2
1
1
0
1
1
1
1
0
0
1
0
insert your data
1
1
0
1
1
1
1
0
0
1
0
highlight the check bits
1
0
1
1
0
0
taking the 1st power of 2^0 (1) check 1 skip 1 = odd parity
1
1
1
1
0
1
taking the 2nd power of 2^1 (2) check 2 skip 2 = odd parity
1
1
1
0
taking the 3rd power of 2^2 (4) check 4 skip 4 = odd parity
1
1
0
1
taking the 4th power of 2^3 (8) check 8 skip 8 = odd parity
All are odd parity, the data should be odd parity, therefore it has been sent and received correctly
00100011110 being sent with even parity
Answer:
11
10
09
08
07
06
05
04
03
02
01
number the columns and highlight the powers of 2
0
0
1
0
0
0
1
1
1
1
0
insert your data
0
0
1
0
0
0
1
1
1
1
0
highlight the check bits
0
1
0
1
1
0
taking the 1st power of 2^0 (1) check 1 skip 1 = odd parity! Dodgy!
0
0
0
0
1
1
taking the 2nd power of 2^1 (2) check 2 skip 2 = even parity. OK
0
0
1
1
taking the 3rd power of 2^2 (4) check 4 skip 4 = even parity! OK
0
0
1
0
taking the 4th power of 2^3 (8) check 8 skip 8 = odd parity. Dodgy!
We have a mixture of odd and even parity, this means that there has been a mistake in sending this data. But where is the error? It has something to do with the lines that have odd parity!
Highlight the column headings that are powers of 2 (1,2,4,8), these are the parity bits
Insert your data and highlight the parity bits
Work your way through the parity bits
2^0 = 1 : check 1, skip 1, check 1, skip 1 ... write down whether it's odd or even parity
2^1 = 2 : check 2, skip 2, check 2, skip 2 ... write down whether it's odd or even parity
2^2 = 4 : check 4, skip 4, check 4, skip 4 ... write down whether it's odd or even parity
etc..
If there is a disparity between rows, highlight all the error data and find where it overlaps
Example: Even Parity Hamming Code Check
Let's take a look a an example of data sent with even parity
11
10
09
08
07
06
05
04
03
02
01
number the columns and highlight the powers of 2
1
0
0
1
0
0
1
0
1
1
1
insert your data
1
0
0
1
0
0
1
0
1
1
1
highlight the check bits
1
0
0
1
1
1
taking the 1st power of 2^0 (1) check 1 skip 1 = even parity
1
0
0
0
1
1
taking the 2nd power of 2^1 (2) check 2 skip 2 = odd parity PROBLEM!
0
0
1
0
taking the 3rd power of 2^2 (4) check 4 skip 4 = odd parity PROBLEM!
1
0
0
1
taking the 4th power of 2^3 (8) check 8 skip 8 = even parity
Note that two of the lines, 2^1 and 2^2, show that an error has been detected. This means that somewhere that these lines cross over a bit has been corrupted, namely bit 6 or bit 7. If we know which one it is we can then switch it and correct the error.
Look at the other checks that are in play, do any of them take part in this crossover? Looking at it, the 2^0 line also checks column 7 and it found it fine. So we are left with column 6 being the problematic one. As Hamming code is corrective, let's flip that column and we should have a correct piece of data.
Another way of finding errors is to add the check digit values together, the error occurs where the check digit equals 4 and 2. Add 4 +2 = 6, the error is with the 6th digit!
11
10
09
08
07
06
05
04
03
02
01
number the columns and highlight the powers of 2
1
0
0
1
0
1
1
0
1
1
1
insert your data
1
0
0
1
0
1
1
0
1
1
1
highlight the check bits
1
0
0
1
1
1
taking the 1st power of 2^0 (1) check 1 skip 1 = even parity
1
0
0
1
1
1
taking the 2nd power of 2^1 (2) check 2 skip 2 = even parity
0
1
1
0
taking the 3rd power of 2^2 (4) check 4 skip 4 = even parity
1
0
0
1
taking the 4th power of 2^3 (8) check 8 skip 8 = even parity
The number is now even parity and correct: 10010110111
Exercise: Detect and Correct the error in the following Hammed Code
Now try this example with even parity:
01101001011
Answer:
11
10
09
08
07
06
05
04
03
02
01
number the columns and highlight the powers of 2
0
1
1
0
1
0
0
1
0
1
1
insert your data
0
1
1
0
1
0
0
1
0
1
1
highlight the check bits
0
1
1
0
0
1
taking the 1st power of 2^0 (1) check 1 skip 1 = odd parity PROBLEM!
0
1
1
0
0
1
taking the 2nd power of 2^1 (2) check 2 skip 2 = odd parity PROBLEM!
1
0
0
1
taking the 3rd power of 2^2 (4) check 4 skip 4 = even parity
0
1
1
0
taking the 4th power of 2^3 (8) check 8 skip 8 = even parity
The error is in the lines crossing over, that is on lines 2^0 and 2^1, but which bit is it?
You'll notice that the 2^2 and 2^3 lines are correct so we can discount any bits that are covered in those lines. This leaves us with the third column. Flipping this value gives us the corrected value of: 01101001111
11111101000 sent with odd parity
Answer:
11
10
09
08
07
06
05
04
03
02
01
number the columns and highlight the powers of 2
1
1
1
1
1
1
0
1
0
0
0
insert your data
1
1
1
1
1
1
0
1
0
0
0
highlight the check bits
1
1
1
0
0
0
taking the 1st power of 2^0 (1) check 1 skip 1 = odd parity
1
1
1
1
0
0
taking the 2nd power of 2^1 (2) check 2 skip 2 = even parity PROBLEM!
1
1
0
1
taking the 3rd power of 2^2 (4) check 4 skip 4 = odd parity
1
1
1
1
taking the 4th power of 2^3 (8) check 8 skip 8 = even parity PROBLEM!
The error is in the lines crossing over, that is on lines 2^1 and 2^3, but which bit is it? We can look at the place they cross over, bit ten, alternatively we can add the parity bit numbers together the row of parity bit 2 plus the row of parity bit: 2 + 8 = bit 10.
Flipping this value gives us the corrected value of: 10111101000
00111000101 sent with even parity
Answer:
11
10
09
08
07
06
05
04
03
02
01
number the columns and highlight the powers of 2
0
0
1
1
1
0
0
0
1
0
1
insert your data
0
0
1
1
1
0
0
0
1
0
1
highlight the check bits
0
1
1
0
1
1
taking the 1st power of 2^0 (1) check 1 skip 1 = even parity
0
0
1
0
1
0
taking the 2nd power of 2^1 (2) check 2 skip 2 = even parity
1
0
0
0
taking the 3rd power of 2^2 (4) check 4 skip 4 = odd parity. PROBLEM!
0
0
1
1
taking the 4th power of 2^3 (8) check 8 skip 8 = even parity
There is only one line with an error, the line of parity bit 4. This means that the error is in bit 4,5,6 or 7. Parity bit lines 1 and 2 imply that bits 5,6,7 are all fine, leaving us with the error in bit 4. Alternatively, as the error only occurs on parity bit line 4, then we know the error is with bit 4!
Flipping this value gives us the corrected value of: 00111001101