1. Error Correction Codes
Can we fix what is broken?
Copyright © by Curt Hill

2. The Story Starts with Parity
In the olden days, well before computers, the teletype existed to transmit telegrams
In transmitted characters over telegraph lines
It replaced to older Morse code keys
This communication form had many options and was error prone
Parity was the first solution
Copyright © by Curt Hill

3. Copyright © 2003-2017 by Curt Hill
Parity
Simple error detection system
Add one parity bit to the data bits, usually at the end
Two forms: even and odd
Even parity
Set the parity bit so that the number of 1 bits is even
Odd parity is similar
Copyright © by Curt Hill

4. Copyright © 2003-2017 by Curt Hill
Example
Consider the following data:
Even parity:
Odd parity:
Even parity:
Odd parity:
Copyright © by Curt Hill

5. Copyright © 2003-2017 by Curt Hill
Options
How many bits were being transmitted in a character?
How fast were they being transmitted?
What parity was being used?
How many start and stop bits?
The answers to all these questions constitute a transmission protocol
The choice of protocol did not matter as both the sender and receiver agreed on the same one
Copyright © by Curt Hill

6. Copyright © 2003-2017 by Curt Hill
Teletype and Parity
Parity will detect a one bit error
Even if it is the parity bit that has the error
Errors then triggered a retransmit
It cannot detect a two bit error
Usually before the two bit errors set in there are several one bit errors
This was satisfactory for applications in the early to middle twentieth century
Copyright © by Curt Hill

7. Is this the best we can do?
If we employ multiple error detection bits we can detect multiple bit errors and correct single bit errors
How do we correct an error?
Since each bit may only have two states 0 or 1 all we have to know is which bit is bad
Correct it by reversing it
How do find location of the error?
Copyright © by Curt Hill

8. Copyright © 2003-2017 by Curt Hill
Basic Scheme
Four data bits (0-3)
Three syndrome bits (a-c)
Each parity bit protects just three of the four
A protects 0-2 with parity
B protects 1-3 with parity
C protects 0,1,3 with parity
Each bit is protected by two or three of the parity bits
The number of parity bits indicate which bit was in error
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9. Copyright © 2003-2017 by Curt Hill
Consider four bits with three error detection bits What happens when any bit is wrong? 2 b a 1 3 c
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10. Copyright © 2003-2017 by Curt Hill
So?
If the parity bit is in error, then only one parity bit is flagged
Can be ignored, well maybe
If a data bit is in error, two or three parity bits will detect it
Error in 0 – a, c
Error in 1 – a, b, c
Error in 2 – a, b
Error in 3 – b, c
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11. Copyright © 2003-2017 by Curt Hill
Multi-bit Errors? 1 b a 2 3 c
If we have a double bit error we can detect but not correct.
Adding more parity bits could correct even this.
Copyright © by Curt Hill

12. Copyright © 2003-2017 by Curt Hill
Is this too expensive?
This almost doubles the number of bits
4 data and 3 check (42%)
Doubling the data bits only adds one more to the syndrome
Thus:
8 data and 4 syndrome bits (33%)
16 data and 5 syndrome bits (23%)
32 data and 6 syndrome bits (15%)
64 data and 7 syndrome bits (10%)
The larger the better
Compare this to keeping a redundant copy
Copyright © by Curt Hill

13. Copyright © 2003-2017 by Curt Hill
Finally
Error correction has been applied to many areas:
Transmission of data
Checking or correcting for errors in memory
Checking or correcting for errors on disk/tape
We covered this because we need it for RAID
Copyright © by Curt Hill