section 3.5: error-correcting codes

Section 3.5: Error-Correcting Codes

Math for Liberal Studies

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

Transmission Error Examples

“Party tonight, bring chipd”



We detect the error because “chipd” is not a word in our dictionary



We detect the error because “chipd” is not a word in our dictionary

Can we correct the error?



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


“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)


“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


“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


“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

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)

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

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


Suppose these messages are 4 digits long

That makes 16 possible messages NASA could send:

0000 0001 0010 00110100 0101 0110 01111000 1001 1010 10111100 1101 1110 1111


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


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

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

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

Adding a Parity Checksum Digit

Let’s go back and add a parity checksum digit to each of these messages

0000 0001 0010 00110100 0101 0110 01111000 1001 1010 10111100 1101 1110 1111


For example: 1011

The sum of the digits is 3, which has parity 1

So the code word is 101110000 0001 0010 00110100 0101 0110 01111000 1001 1010 10111100 1101 1110 1111


Doing this for each of the messages gives us the code words shown below

00000 00011 00101 0011001001 01010 01100 0111110001 10010 10100 1011111000 11011 11101 11110

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

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

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

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

Using Distance

We compare the message we received (00100) to the valid code words:

Code Word Distance Code

Word Distance Code Word Distance Code

Word Distance

00000 1 00011 3 00101 1 00110 101001 3 01010 3 01100 1 01111 310001 3 10010 3 10100 1 10111 311000 3 11011 5 11101 3 11110 3

Using Distance

Unfortunately, there are 5 code words that are tied for the closest

We have no way of knowing which one is correct!

Code Word Distance Code

Word Distance Code Word Distance Code

Word Distance

00000 1 00011 3 00101 1 00110 101001 3 01010 3 01100 1 01111 310001 3 10010 3 10100 1 10111 311000 3 11011 5 11101 3 11110 3

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

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

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

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

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



Let’s try it on an example: 0111

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

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:

0000000 0001011 0010111 00111000100101 0101110 0110010 01110011000110 1001101 1010001 10110101100011 1101000 1110100 1111111

Minimum Distance

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

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

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

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

section 3.5: error-correcting codes

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