Section 3.5: Error-Correcting Codes

1. Math for Liberal Studies Section 3.5: Error-Correcting Codes

2. Transmission Problems • Problems can occur when data is transmitted from one place to another • The two main problems are • transmission errors: the message sent is not the same as the message received • security: someone other than the intended recipient receives the message

3. Transmission Error Examples • “Party tonight, bring chipd”

4. Transmission Error Examples • “Party tonight, bring chipd” • We detect the error because “chipd” is not a word in our dictionary

5. Transmission Error Examples • “Party tonight, bring chipd” • We detect the error because “chipd” is not a word in our dictionary • Can we correct the error?

6. Transmission Error Examples • “Party tonight, bring chipd” • Even though “chipd” is not the correct word, we can assume that the correct word is close • What words are one letter away from “chipd”? • After considering the possibilities, “chips” is the most likely correction

7. Transmission Error Examples • “Party tonight, bring sofa” • This time, all of the words are in the dictionary, but we still suspect something is wrong (unless it’s a furniture party)

8. Transmission Error Examples • “Party tonight, bring sofa” • This time, all of the words are in the dictionary, but we still suspect something is wrong (unless it’s a furniture party) • Again we can change a single letter to change “sofa” to “soda,” which seems likely to be the original intended message

9. Transmission Error Examples • “Party tonight, bring sedr” • Identifying the error is easy: “sedr” is not a word • However, this time, changing a single letter doesn’t get us a word that makes sense

10. Transmission Error Examples • “Party tonight, bring sedr” • We can change two letters, but that gives us two viable options: • sedr → sodr → soda • sedr → bedr → beer • It is impossible to tell which of these was the original intended message

11. Error Correction Principles • Errors can be detected when the message isn’t in a “dictionary” of valid messages • We can try to correct errors by finding valid messages that are “close” to the message we receive (but this doesn’t always work)

12. Digital Languages • Machines communicate with each other using a language entirely made of 0’s and 1’s • The same kinds of errors we studied earlier (substitution, transposition) can occur when these digital signals are sent • We can use special techniques to detect and correct these errors

13. Sending Signals to Mars • As an example, consider the Mars rovers, which landed in 2004 • NASA sends signals to the rover to command it to perform various tasks, like movement • These signals are sent in binary

14. Sending Signals to Mars • Suppose these messages are 4 digits long • That makes 16 possible messages NASA could send:

15. Sending Signals to Mars • Suppose NASA sends the message “0110,” which mightbe telling the rover to move backwards to avoid a crater • If, over the vast distances between planets, the message is garbled and received as “0010,” this could be disastrous

16. Sending Signals to Mars • If the garbled message is interpreted as “move forward,”this could mean the end of avery expensive mission • To avoid this problem, we will add check digits to the message, just like we did for ID numbers

17. Parity Checksums • Many of the check digit schemes we studied involved adding up the digits of our ID number • We’ll do something similar here, but keep in mind that since every digit of a binary message must be 0 or 1, our check digit must be 0 or 1 also

18. Parity Checksums • A “checksum” is just a check digit that is based on a sum of digits in the message • The “parity” of the sum is 0 if the sum is even, and 1 if the sum is odd • Another way to think about parity is that it is the remainder when the sum is divided by 2

19. Adding a Parity Checksum Digit • Let’s go back and add a parity checksum digit to each of these messages

20. Adding a Parity Checksum Digit • For example: 1011 • The sum of the digits is 3, which has parity 1 • So the code word is 10111

21. Adding a Parity Checksum Digit • Doing this for each of the messages gives us the code words shown below

22. Testing the New System • Now when NASA wants to send the message “0110,” they send the code word “01100.” • Now see what happens when there is a substitution error: 00100 • We can detect the error because this is not a valid code word

23. Testing the New System • Can we correct the error? • Using the ideas from before, we want to look for the valid code word that is “closest” to the message we received • What does “closest” mean? We have to define the idea of distance between code words

24. Distance • The distance between two code words is simply the number of digits in which they differ • For example, the distance between 01101 and 10111 is 3

25. Using Distance • To correct the error in our message, we will compare it to every valid message and find the one that is closest (in the sense of having the smallest distance) • This is called the minimum distance decoding method

26. Using Distance • We compare the message we received (00100) to the valid code words:

27. Using Distance • Unfortunately, there are 5 code words that are tied for the closest • We have no way of knowing which one is correct!

28. What Went Wrong? • Why didn’t our checksum allow us to correct this error? • If we look closely at our list of code words, we see that some of them are at a distance of 2 from each other

29. What Went Wrong? • Distance 2 is significant because it means that if there is a single error, the new message is now 1 away from the original, but also 1 away from a new code word

30. What Went Wrong? • If we can create a code system where the minimum distance between code words is 3, then we will be able to correct any single digit error

31. More Checksums! • Our solution is to add more checksums to our messages • Let’s call the four digits of our message M1, M2, M3, and M4 • So for the message 0110, M1 = 0, M2 = 1, M3 = 1, and M4 = 0

32. More Checksums! • This time we will have three checksums, which we’ll call C1, C2, and C3 • C1 is the parity of M1 + M2 + M3 • C2 is the parity of M1 + M3 + M4 • C3 is the parity of M2 + M3 + M4 • Let’s try it on an example: 0111

33. More Checksums! • Our message is 0111 • C1 is the parity of M1 + M2 + M3 = 2, which is 0 • C2 is the parity of M1 + M3 + M4 = 2, which is 0 • C3 is the parity of M2 + M3 + M4 = 3, which is 1 • So the code word is 0111001

34. A New List of Code Words • Doing this for each of our 4-digit messages, we get a new list of 7-digit code words:

35. Minimum Distance • This time, the minimum distance between code words is 3, which means that we can detect any single error

36. Minimum Distance • If we start with a valid code word and there is a single error, we are 1 away from where we started, and at least 2 away from anywhere else

37. Minimum Distance • Also, we can detect any two errors using this code, since after 2 errors, we are still at least 1 away from any valid code word

38. Using Minimum Distance • In general, if we know that the minimum distance between code words is D: • the code can detect D – 1 errors • the code can correct (D – 1)/2 errors, rounded down • In our examples, when D = 2, we could detect 1 error, but could not correct any • When D = 3, we can detect 2 errors, can correct 1