Math for Liberal Studies

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)

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

Mostly, whenever information needs to be transmitted from one location to another, a binary code is used

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

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”?

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 11000001 0101 1001 11010010 0110 1010 11100011 0111 1011 1111

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

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

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 circlecontains an even number ofones

IIIIVII

I

VII

V VI

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 0100101

001

0

1

1 0

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: that’s digit IV

Since we know digit IV is wrong, there isonly one way to fix it: change it from 1 to 0

011

0

1

1 0

Encode the message 1110 using this method

You have received the message 0011101. Find and correct the error in this message.

Binary codes can be used to represent more conventional information, but 4 digits only gives us 16 possible messages

That’s not even enough to represent the alphabet!

If we have n digits, then we can make 2n different messages

5 digits -> 32 messages 6 digits -> 64 messages, etc.

The idea we’re using is a specific example of a parity check sum

The parity of a number is either odd or even For example, digit V is 0 if I + II + III is even,

and odd if I + II + III is odd

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.

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

Under this new way of thinking about our system, how do we decode messages?

Simply compare the message with the list of possible correct messages and pick the “closest” one

What should “closest” mean? If you have two messages of the same length,

the distance between the two messages is the number of digits in which they differ

What is the distance between 1100101 and 1010101? 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

The nearest neighbor decoding method decodes a received message as the code word that agrees with the message in the most positions

In this example, our messages are three digits long: a1a2a3

We have three check digits c1 = 0 if a1 + a2 + a3 is even c1 = 1 if a1 + a2 + a3 is odd c2 = 0 if a1 + a3 is even c2 = 1 if a1 + a3 is odd c3 = 0 if a2 + a3 is even c3 = 1 if a2 + a3 is odd

Using these rules, we can find all of our code words

By analyzing this list,we see that the smallestdistance between twocode words is 3

That means we can usethese code words to either detect two errorsor correct one error

Message Code Word

000 000000

001 001111

010 010101

011 011010

100 100110

101 101001

110 110011

111 111100