error correcting codes_oon

7/29/2019 Error Correcting Codes_oon

1/37

Error correcting codes

Oon Boey Lay

Rujukan: EMM400 StudyGuideFAEMM400Tasks&Solns1


2/37

The need for powerful (and public) encryption

has grown out of our ever increasing reliance

on digital information you will no doubt

have seen yourselves how when you

sometimes go to or leave a web page you are

told that the page is (or is not) encrypted for

security purposes. Security issues arise notonly in the public domain but also in the

secrecy that surrounds military operations.


3/37

There is a conflict between governments

trying to restrict the use of encryption which

they regard as a tool for terrorists, and civil

libertarians who argue that encryption is

necessary to protect privacy


4/37

Another product of the digital age is error-

correcting codes. This area of mathematics

began just 60 years ago in 1948.


5/37

Have you ever wondered how satellites send

photographs back to earth? Or why you cant

just invent a credit card number when you try

to buy something on the internet? Or how you

can be sure when you deposit money into an

ATM that the correct amount will go to your

account?


6/37

We will begin by looking at:

what are error-correcting codes and why we

need them;

the Mariner 6 space mission; and

some examples of error-correcting codes.


7/37

The Mariner 6 space mission

In 1965, the USAs Mariner 4 was the first

spaceship to photograph Mars. At that time,

each picture took 8 hours to transmit. Later

Mariner missions, such as Mariner 6,

produced improved pictures, using what are

called error-correcting codes.


8/37

Mariner 6 spaceship sending pictures from Mars


9/37

FIGURE 2.2

Converting Mariner 6

pictures to binary digits

In order to transmit pictures, a fine grid was placed on the

picture and each square or what we would now refer to

as a pixel, was given a degree of blackness ranging from 0 to 63


10/37

0 0000001 000001

2 000010

3 000011

4

0001005 000101

6 000110

7 000111

8 001000

9 001001

43 101011

63

111111

Each of these numbers was written as a sequence of six 0s and

1s, for example by writing it in the binary system (that is, in

base 2), as shown below.


11/37

So we have degree of blackness = 43

101011.

In the case of Mariner 6, each picture was

broken up into 700 x 832 squares, so if each

square was coded using 6 binary digits, each

picture would consist of a sequence of 6 x 700

x 832 = 3 494 400 binary digits.


12/37

However, while the degree of blackness of

each square consisted of a six binary digits,

the message that was sent actually used many

more digits for each degree of blackness in

fact 32 binary digits were sent for each

square, so each picture consisted of a

sequence of 32 x 700 x 832 = 18 636 800binary digits.


13/37

Message to

betransmitted

Coder

Binary code

to betransmitted

Decoder

Binarycode

received

Channel

transmittingcode

Messagereceived

Transmitting data through a communication channel


14/37

Errors introduced by noise


15/37

If we have many errors like this they will affectthe quality of the picture that is received andwe cannot really ask for the message to be

repeated. This is the reason we need error-correcting

codes.

Error-correcting codes combat error byintroducing redundancy by including moresymbols than is necessary for the message.


16/37

Digital alphabet

The process of encoding a message usuallybegins with the conversion of ordinary textinto a string of numbers by means of a digitalalphabet like the one shown below. In the

code shown in Figure 2.6, each letter (and afew punctuation marks) are represented by asequence of 0s and 1s of length 5 we canthink of these sequences as being thenumbers between 0 and 32 written in thebinary (base 2) system.


17/37

= 00000A = 00001 B = 00010 C = 00011 D = 00100

E = 00101 F = 00110 G = 00111 H = 01000 I = 01001

J = 01010 K = 01011 L = 01100M = 01101N = 01110

O = 01111 P = 10000 Q = 10001 R = 10010 S = 10011

T = 10100 U = 10101V = 10110W = 10111 X = 11000

Y =

110001

Z = 11010 , = 11011 . = 11100 ? = 11101

: = 11110 ; = 11111

A binary code using the numbers 0 to 32 in base 2


18/37

So, for example, if I were to write my name in

this code I would get:

SUSIE 10011 10101 10011 01001 00101

ACTIVITY 2.1

Write your name using the binary code

above.


19/37

An example of a single-error-correcting code

In this example, we will assume that everymessage we want to transmit consists of just 3binary digits. There are 8 such possible messages,

so we could think of them as representing theintegers from 0 to 7.

In this example, we will add an extra 5 redundantdigits to each message as shown below to give

codewords oflength 8. (Right now we wont tryto explain how we choose the extra digits.)


20/37

0 = 000 000 00000

1 = 001 001 10110

2 = 010 010 10101

3 = 011 011 00011

4 = 100 100 10011

5 = 101 101 00101

6 = 110 110 00110

7 = 111 111 10000

A single-error-correcting code of length 8


21/37

ACTIVITY 2.2

Check that every codeword differs

from every other codeword inexactly four places.


22/37

Imagine that we receive the following

message: 00111110

we see that the codeword closest to 00111110

is 00110110.

It differs in just one place the fifth place

(underlined here).

hence the name single-error-correcting code.


23/37

In this example, we have:

8 digit codewords with

3 information digits and 5 redundant digits.

So we can say this code has information rate

= 3/8


24/37

Repetition codes

One simple way of introducing redundancy is

to repeat everything. So if we have a message

we could decide to code it by repeating each

digit n times.

For example, ifn = 5, we get a repetition code

oflength 5.


25/37

S 10011 11111 00000 00000 11111 11111

U 10101 11111 00000 11111 00000 11111

S 10011 11111 00000 00000 11111 11111

I 01001 00000 11111 00000 00000 11111

E 00101 00000 00000 11111 00000 11111

In my example earlier, my name would be coded as

shown below.


26/37

For example what should we do if we receive

the following message:

11011 00110 11000 10000 10111 ?


27/37

A decoding algorithm for a repetition code of length 5

The following is an example of a decoding

algorithm for any one 5-digit block:

Count the number of 1s.

If number of 1s 3 , write 11111.

If number of 1s 2 , write 00000.


28/37

00000 10010 11011 11000 01111

11110 01010 01000 01011 00001

00111 10000 01100 11100 00000

01000 11111 00111 10111 11101

01111 00010 01000 10111 10000

ACTIVITY 2.3

You receive the following message, which was sent

using the alphabet from Activity 2.1 and a repetitioncode of length 5.


29/37

Convert the message to the 5 digit binary

code.

Use the alphabet from Activity 2.1 to convert

it to ordinary letters.


30/37

00000 11011 01111 00100 11111

01000 11111 00100 00001 11101

11101 00100 00000 11110 01111

11111 10000 01000 00000 00100

10111 00010 10000 00111 01100

01100 10001 01010 00011 11001

01010 10111 01111 11111 00000

11111 00010 11011 01000 00000

ACTIVITY 2.4

Using the same code as in Activity 2.3, you receive the

following message:


31/37

Convert the message to the 5 digit binary

code.

Use the alphabet from Figure 2.6 to convert it

to ordinary letters.

What message do you think was really

intended?


32/37

Single parity-check codes

Single parity-check codes are the other

extreme from repetition codes. While

repetition codes have a single message digit,

single parity-check codes have only one checkdigit.

The check digit is obtained by taking the sum

of the information digits mod 2.


33/37

A 000001 1

5 messagedigits

1check digit

B 000010 1

C 000011 0

D 000100 1

So, for example, using the code in Figure 2.6, we get


34/37

In general, the codewords look like c1 c2 c3 c4

c5 c6 where

c6 = c1 +c2 +c3 +c4 + c5(mod 2).


35/37

ACTIVITY 2.8

Find the codewords representing each of the following letters in thesingle-parity-check code above: J , L , Q , S , G , X.

ACTIVITY 2.9

Write the message NO ERRORS in the single-parity-check code above.

ACTIVITY 2.10

The following message is received in the single-parity-check code:

000011 000000 001111 011110 010110 001001000000 100100 001010 100111 101001 011000

101000 Detect the places where the errors occur.

Decode the remaining letter.

Use guesswork to complete the message.


36/37

The information rate is =

which is very high!

However, a single-parity-check code can only

detectan odd number of errors and cannot

correct them!

n

n 1

n

k

SINGLE-PARITY- CHECK CODE


37/37

error correcting codes_oon

Documents