MAT 1000
Mathematics in Today's World
Winter 2015
Last Time
Identification numbers often include check digits: extra
digits that allow us to catch errors.
There are several different methods for finding check digits
used in practice.
We looked at sytems which are used for UPC codes, credit
card numbers, and ISBNs as well as bar codes.
Today
Binary codes are strings consisting of either 0s
or 1s.
We will look at a specific way to encode binary
messages using Venn diagrams
Then we consider a more general method,
called “parity check sums”
Binary codes
Binary codes are messages which are
represented using only the digits 0 and 1.
Some examples of binary codes of length three
are 101, or 110, or 000
Computers use binary codes internally.
Binary codes
We can append extra digits to binary codes to
help catch errors.
In fact, we can either identify where the error
occurs, or we can fix the error.
This requires appending more than a single
digit.
Binary codes
One advantage to binary codes: each position
is either a 1 or a 0, so there are only two
possible errors:
0 can be received as 1
1 can be received as 0
Venn diagram encoding
If our binary codes have length 4, there is an
encoding/decoding system which uses Venn diagrams.
Use three circles in the following configuration:
Venn diagram encoding
Note that we have four sections of overlap:
Our four digit binary message will be placed in these 4
spaces, in this order.
Then we append three digits, based on the other 3 spaces.
Venn diagram encoding
Example
Let’s encode the message 1011
First, fill in the numbered spaces in the Venn diagram using
the digits of the message, in order: 1st digit in the 1st space,
2nd digit in the second space, and so on.
Venn diagram encoding
Example
We have three more spaces to fill:
We will put either a 1 or a 0 in each of these.
Venn diagram encoding
Example
To decide whether to put in a 0 or a 1, we choose
whichever makes the total number of 1s in each circle
even:
1
0
0
Adding these three extra digits (in order) gives 1011 010
Venn diagram encoding
Example
Now we will see how to correct errors using this method.
Suppose our message 1011 010 is mistakenly received
as 1010 010
When we fill in the Venn diagram, we can see there is a
mistake because some of the circles have an odd number
of 1s in them.
Venn diagram encoding
Example
The upper left circle has an even number of 1s:
Venn diagram encoding
Example
But the other two circles both have an odd number of 1s:
Venn diagram encoding
Example
But the other two circles both have an odd number of 1s:
Venn diagram encoding
Example
Which digit is incorrect?
The incorrect digit must appear in both of the circles with
an odd number of 1s, but it is not in the circle with the
correct number of 1s.
This tells us exactly where the error must be:
Venn diagram encoding
Example
Moreover, once we know the location of the error, we can
fix it.
After all, this is a binary code: if 0 is not the correct digit,
then 1 must be.
Fixing the mistake recovers our original message: 1011
010
Venn diagram encoding
Some disadvantages of this method:
• Only works on messages of length four
• If there are two or more errors, they may go
undetected, or they may be fixed incorrectly
Let’s see an example that shows how two errors can be
fixed incorrectly.
Venn diagram encoding
Example
The original message is 0110. Encode this message.
1
0
0
So the encoded message is: 0110 010
Venn diagram encoding
Example
Suppose the message is received with two errors as
0100 011
What happens when we decode this message?
Venn diagram encoding
Example
The two upper circles have an odd number of 1s in them,
but the lower circle has an even number of 1s. So the
method tells us there is an error in this place:
Venn diagram encoding
Example
This gives the “corrected” message 1100 011
Of course this is not actually the correct message. That
was 0110 010
So this method may fail when there are two or more
errors in the message.
Parity Check Sums
Here are some possible improvements on the Venn
diagram method:
1. Longer messages
2. Correct more errors
To describe these improved methods, we need to look at
the Venn diagram method in a different way, using “parity
check sums”
Parity Check Sums
The parity of a number is whether it is even or odd.
Suppose we have a four-digit binary message 𝑎1 𝑎2 𝑎3 𝑎4
We decide to append three digits, 𝑐1 , 𝑐2 , 𝑐3
We choose 𝑐1 to have the same parity as the sum
𝑎1 + 𝑎2 + 𝑎3
This means that
If 𝑎1 + 𝑎2 + 𝑎3 is even, 𝑐1 = 0
If 𝑎1 + 𝑎2 + 𝑎3 is odd, 𝑐1 = 1
Parity Check Sums
We choose 𝑐2 to have the same parity as the sum
𝑎1 + 𝑎3 + 𝑎4
And 𝑐3 will have the same parity as the sum
𝑎2 + 𝑎3 + 𝑎4
Parity Check Sums
Example
Use these parity check sums to encode the message
1011.
The first check sum is 𝑎1 +𝑎2 + 𝑎3 which is 1 + 0 + 1 = 2.
So the first appended digit is 0
The next check sum is 𝑎1 +𝑎3 + 𝑎4 which is 1 + 1 + 1 = 3.
So the second appended digit is 1
The last check sum is 𝑎2 + 𝑎3 + 𝑎4 which is 0 + 1 + 1 = 2.
So the third appended digit is 0
This makes the message: 1011 010
Parity Check Sums
If you compare this method with the Venn diagram
method we used earlier, you will see that they are
identical (for any four digit message they give the same
code)
The advantage of parity check sums over Venn diagrams
is that we have more flexibility:
• we can now work with longer messages
• we can add more digits (which can catch more errors)
Parity Check Sums
Example
In addition to the digits 𝑐1 , 𝑐2 , 𝑐3 defined above, we could
add another digit 𝑐4 using the sum 𝑎1 + 𝑎2 + 𝑎4 (which
was not used in the Venn diagram method)
Parity Check Sums
Example
For a message of length five 𝑎1 𝑎2 𝑎3 𝑎4 𝑎5 we could use
check digits 𝑐1 , 𝑐2 , 𝑐3 , 𝑐4 which have the same parity as
𝑎1 + 𝑎2 + 𝑎3 + 𝑎4
𝑎2 + 𝑎3 + 𝑎5
𝑎1 + 𝑎2 + 𝑎4
𝑎1 + 𝑎3
There are lots of other possibilities.
Parity Check Sums
If we encode a message with parity check sums, how
should we decode it?
The method used is called “nearest neighbor” decoding.
To use this, we have to discuss the “distance” between
binary strings.
Parity Check Sums
The distance between two binary strings is the number of
places in which they differ.
Example: 101010 and 111010 have a distance of 1
Example: 101010 and 011011 have a distance of 3
Example: 101010 and 101010 have a distance of 0
Note if two strings have different lengths, it doesn’t make
sense to talk about their distance.
The distance between 101010 and 11101 doesn’t make
sense
Parity Check Sums
Nearest neighbor decoding:
Receive an encoded message, which may have some
errors.
Find the nearest correct message (meaning the one
which is the smallest distance from the received
message).
Do not decode if there is a tie.
Parity Check Sums
In order to use nearest neighbor decoding, we need to
make a list of every possible correct message.
In the next example, we will take a parity check sum
method, list every possible correct message, and then
use that list to decode a message.
Parity Check Sums
Example
Messages will be of length three 𝑎1 𝑎2 𝑎3 . We will add three parity sum
digits using the sums 𝑎1 + 𝑎2 + 𝑎3 and 𝑎1 + 𝑎3 and 𝑎2 + 𝑎3
Messages
𝑎1 + 𝑎2 + 𝑎3
𝑎1 + 𝑎3
𝑎2 + 𝑎3
Coded Messages
000
0
0
0
000 000
001
1
1
1
001 111
010
1
0
1
010 101
100
1
1
0
100 110
110
2
1
1
110 011
101
2
2
1
101 001
011
2
1
2
011 010
111
3
2
2
111 100
Parity Check Sums
Example
Decode the message 000101.
We will find the distance from this message to each valid message:
Coded Messages
Distance
000 000
2
001 111
2
010 101
1
100 110
3
110 011
4
101 001
3
011 010
5
111 100
4
The message is decoded to be 010 101
Parity Check Sums
We have lots of choice for different parity check sums.
How many should we use?
Which ones should we use?
We will see next time how to analyze different choices of
parity check sums.