1. MAT 105 Spring 2008
Chapter 17: Binary Codes

2. Binary Codes
A binary code is a system for encoding data made up of 0’s and 1’s
Examples
Postnet (tall = 1, short = 0)
UPC (dark = 1, light = 0)
Morse code (dash = 1, dot = 0)
Braille (raised bump = 1, flat surface = 0)
Movie ratings (thumbs up = 1, thumbs down = 0)

3. Binary Codes are Everywhere
CD, MP3, and DVD players, digital TV, cell phones, the Internet, space probes, etc. all represent data as strings of 0’s and 1’s rather than digits 0-9 and letters A-Z
Whenever information needs to be digitally transmitted from one location to another, a binary code is used

4. Transmission Problems
What are some problems that 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

5. Transmission Error Example
Suppose you were looking at a newspaper ad for a job, and you see the sentence “must have bive years experience”
We detect the error since we know that “bive” is not a word
Can we correct the error?
Why is “five” a more likely correction than “three”?
Why is “five” a more likely correction than “nine”?

6. Another Example
Suppose NASA is directing one of the Mars rovers by telling it which crater to investigate
There are 16 possible signals that NASA could send, and each signal represents a different command
NASA uses a 4-digit binary code to represent this information
0000
0100
1000
1100
0001
0101
1001
1101
0010
0110
1010
1110
0011
0111
1011
1111

7. Lost in Transmission
The problem with this method is that if there is a single digit error, there is no way that the rover could detect or correct the error
If the message sent was “0100” but the rover receives “1100”, the rover will never know a mistake has occurred
This kind of error – called “noise” – occurs all the time

8. Adding Check Digits
One way to try to avoid these errors is to send the same message twice
This would allow the rover to detect the error, but not correct it (since it has no way of knowing if the error occurs in the first copy of the message or the second)
There is a better way to allow the rover to detect and correct these errors, and only requires 3 additional digits

9. Computing the Check Digits
The original message is four digits long
We will call these digits I, II, III, and IV
We will add three new digits, V, VI, and VII
Draw three intersecting circles as shown here
Digits V, VI, and VII should be chosen so that each circle contains an even number of ones V I VI
III IV II
VII

10. Appending Digits to the Message
The message we want to send is “0100”
Digit V should be 1 so that the first circle has two ones
Digit VI should be 0 so that the second circle has zero ones (zero is even!)
Digit VII should be 1 so that the last circle has two ones
Our message is now 1 1 1

11. Detecting and Correcting Errors
Now watch what happens when there is a single digit error
We transmit the message and the rover receives
The rover can tell that the second and third circles have odd numbers of ones, but the first circle is correct
So the error must be in the digit that is in the second and third circles, but not the first: that’s digit IV
Since we know digit IV is wrong, there is only one way to fix it: change it from 1 to 0 1 1 1 1

12. Try It! Encode the message 1110 using this method
You have received the message Find and correct the error in this message.

13. Extending This Idea
This method only allows us to encode 16 possible messages, which isn’t even enough to represent the alphabet!
However, if we use more digits, we won’t be able to use the circle method to detect and correct errors
We’ll have to come up with a different method that allows for more digits

14. Parity Check Sums
The circle method is a specific example of a “parity check sum”
The “parity” of a number is 1 is the number is odd and 0 if the number is even
For example, digit V is 0 if I + II + III is even, and 1 if I + II + III is odd

15. Conventional Notation
Instead of using Roman numerals, we’ll use a1 to represent the first digit of the message, a2 to represent the second digit, and so on
We’ll use c1 to represent the first check digit, c2 to represent the second, etc.

16. Old Rules in the New Notation
Using this notation, our rules for our check digits become
c1 = 0 if a1 + a2 + a3 is even
c1 = 1 if a1 + a2 + a3 is odd
c2 = 0 if a1 + a3 + a4 is even
c2 = 1 if a1 + a3 + a4 is odd
c3 = 0 if a2 + a3 + a4 is even
c3 = 1 if a2 + a3 + a4 is odd c1 a1 c2 a3 a4 a2 c3

17. An Alternative System
If we want to have a system that has enough code words for the entire alphabet, we need to have 5 message digits: a1, a2, a3, a4, a5
This gives us 32 possible messages… enough for the alphabet and then some
We will also need more check digits to help us decode our message: c1, c2, c3, c4

18. Rules for the New System
We can’t use the circles to determine the check digits for our new system (if we tried to draw 5 circles, things would get very hard), so we use the parity notation from before
c1 is the parity of a1 + a2 + a3 + a4
c2 is the parity of a2 + a3 + a4 + a5
c3 is the parity of a1 + a2 + a4 + a5
c4 is the parity of a1 + a2 + a3 + a5

19. An Example Code Word
What check digits do we have to add on to the message “00101”? This message will represent the letter “F”
c1 is the parity of a1 + a2 + a3 + a4 = 1, which is 1
c2 is the parity of a2 + a3 + a4 + a5 = 2, which is 0
c3 is the parity of a1 + a2 + a4 + a5 = 1, which is 1
c4 is the parity of a1 + a2 + a3 + a5 = 2, which is 0
So the code word will be

20. Making the Code
Using 5 digits in our message gives us 32 possible messages, we’ll use the first 26 to represent letters of the alphabet
On the next slide you’ll see the code itself, each letter together with the 9 digit code representing it

21. The Code Letter Code A 000000000 N 011010101 B 000010111 O 011101100 C P D Q E R F S G T H U I V J W K X L Y M Z

22. Using the Code Now that we have our code, using it is simple
When we receive a message, we simply look it up on the table
But what happens when the message we receive isn’t on the list?
Then we know an error has occurred, but how do we fix it? We can’t use the circle method anymore

23. Beyond Circles Using this new system, how do we decode messages?
Simply compare the (incorrect) message with the list of possible correct messages and pick the “closest” one
What should “closest” mean?
The distance between the two messages is the number of digits in which they differ

24. The Distance Between Messages
What is the distance between and ?
The messages differ in the 2nd and 3rd digits, so the distance is 2
What is the distance between and ?
The messages differ in all but the 7th digit, so the distance is 6

25. Nearest-Neighbor Decoding
The nearest neighbor decoding method decodes a received message as the code word that agrees with the message in the most positions
Another way to think of it is that the “nearest neighbor” is the message that is the smallest distance away from the message we receive

26. Trying it Out
Suppose that, using our alphabet code, we receive the message
We can check and see that this message is not on our list
How far away is it from the messages on our list?

27. Distances From 010100011 Code Distance 000000000 4 011010101 5 3 6 8 2 1

28. Fixing the Error
Since was closest to the message that we received, we know that this is the most likely actual transmission
We can look this corrected message up in our table and see that the transmitted message was (probably) “K”
This might still be incorrect, but other errors can be corrected using context clues or check digits