Chapter 17: Binary Codes

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Chapter 17 Binary Codes

• MAT 105 Spring 2008

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Binary Codes

• A binary code is a system for encoding data made
  up of 0s and 1s
• 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)

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Binary Codes are Everywhere

• CD, MP3, and DVD players, digital TV, cell
  phones, the Internet, space probes, etc. all
  represent data as strings of 0s and 1s 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

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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

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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?

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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
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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

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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

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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 bechosen so that
  each circlecontains an even number ofones

I
V
VI
III
IV
II
VII
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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 thatthe last circle has
  two ones
• Our message is now 0100101

0
1
0
0
0
1
1
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Detecting and Correcting Errors

• Now watch what happens when there is a single
  digit error
• We transmit the message 0100101 and the rover
  receives 0101101
• 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
  thats digit IV
• Since we know digit IV is wrong, there isonly
  one way to fix it change it from 1 to 0

0
1
0
0
1
1
1
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Try It!

• Encode the message 1110 using this method
• You have received the message 0011101. Find and
  correct the error in this message.

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Extending This Idea

• This method only allows us to encode 16 possible
  messages, which isnt even enough to represent
  the alphabet!
• However, if we use more digits, we wont be able
  to use the circle method to detect and correct
  errors
• Well have to come up with a different method
  that allows for more digits

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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

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Conventional Notation

• Instead of using Roman numerals, well use a1 to
  represent the first digit of the message, a2 to
  represent the second digit, and so on
• Well use c1 to represent the first check digit,
  c2 to represent the second, etc.

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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

a1
c1
c2
a3
a4
a2
c3
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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

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Rules for the New System

• We cant 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

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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 001011010

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Making the Code

• Using 5 digits in our message gives us 32
  possible messages, well use the first 26 to
  represent letters of the alphabet
• On the next slide youll see the code itself,
  each letter together with the 9 digit code
  representing it

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The Code
Letter Code Letter Code
A 000000000 N 011010101
B 000010111 O 011101100
C 000101110 P 011111011
D 000111001 Q 100001011
E 001001101 R 100011100
F 001011010 S 100100101
G 001100011 T 100110010
H 001110100 U 101000110
I 010001111 V 101010001
J 010011000 W 101101000
K 010100001 X 101111111
L 010110110 Y 110000100
M 011000010 Z 110010011
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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
  isnt on the list?
• Then we know an error has occurred, but how do we
  fix it? We cant use the circle method anymore

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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

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The Distance Between Messages

• What is the distance between 110010110 and
  101010110?
• The messages differ in the 2nd and 3rd digits, so
  the distance is 2
• What is the distance between 1110010 and 0001100?
  
• The messages differ in all but the 7th digit, so
  the distance is 6

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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

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Trying it Out

• Suppose that, using our alphabet code, we receive
  the message 010100011
• We can check and see that this message is not on
  our list
• How far away is it from the messages on our list?

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Distances From 010100011
Code Distance Code Distance
000000000 4 011010101 5
000010111 4 011101100 5
000101110 4 011111011 3
000111001 4 100001011 4
001001101 6 100011100 8
001011010 6 100100101 4
001100011 2 100110010 4
001110100 6 101000110 6
010001111 3 101010001 6
010011000 5 101101000 6
010100001 1 101111111 6
010110110 3 110000100 5
011000010 3 110010011 3
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Fixing the Error

• Since 010100001 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