By Gábor P. Nagy, University of Szeged Bolyai Institute.
An Alphabet is a finite set of symbols. We want to transmit a text message, written with the given alphabet, between digital devices.
During transmission, a noise causes that the sent and received sequence of symbols differ.
Our goal is to find methods which are able to correct high percentage of the occuring errors.
[ aábcdeéfghiíjklmnoóöőpqrstuúüűvwxyzAÁBCDEÉFGHIÍJKLMNOÓÖŐPQRTSUÚÜŰVWXYZ
0123456789.,:;?!()+-*/=€$'******************************]
The number of admitted letters: .
| Plain text | Digitalized text |
|---|---|
John Lennon: Imagine Imagine there's no heaven It's easy if you try No hell below us Above us only sky Imagine all the people Living for today... Imagine there's no countries It isn't hard to do Nothing to kill or die for And no religion too Imagine all the people Living life in peace... You may say I'm a dreamer But I'm not the only one I hope someday you'll join us And the world will be as one Imagine no possessions I wonder if you can No need for greed or hunger A brotherhood of man Imagine all the people Sharing all the world... You may say I'm a dreamer But I'm not the only one I hope someday you'll join us And the world will live as one |
Length of the plain text message: . Bit length of the digitalized text: .
Ratio of bits changed by the noise: 0%. Ratio of letters changed by the noise: 0%.
Can you explain this?
| Noisy plain text | Noisy digitalized text |
|---|---|
|
|
The trick: Repeat every bit three times!
The price: You have to transmit a three times longer binary sequence.
| The parameters of the repetition code | |
| Length of the code: | 0 |
| Dimension of the code: | 0 |
| The generator matrix: |
1
|
| Digitalized text | Digitalized text encoded with repetition code |
|---|---|
Ratio of bits changed by the noise: 0%. Ratio of letters changed by the noise: 0%.
| Noisy encoded digital text | Error corrected noisy encoded text |
|---|---|
|
|
Can we do that better? YES! Use the Hamming code.
The price: You have to transmit a 7/4-times longer binary sequence.
| The parameters of the repetition code | |
| Length of the code: | 0 |
| Dimension of the code: | 0 |
| The generator matrix: |
1000
|
| Digitalized text | Digitalized text encoded with Hamming code |
|---|---|
Ratio of bits changed by the noise: 0%. Ratio of letters changed by the noise: 0%.
While the threefold repetition code corrects 1 error per codeword (of length 3), the Golay code corrects 1 errors per codewords (of length 7).
| Noisy encoded digital text | Error corrected noisy encoded text |
|---|---|
|
|
Can we do that better? YES! Use the (extended) Golay code.
The price: You have to transmit a twice longer binary sequence.
| The parameters of the repetition code | |
| Length of the code: | 0 |
| Dimension of the code: | 0 |
| The generator matrix: |
100000000000
|
| Digitalized text | Digitalized text encoded with extended Golay code |
|---|---|
Ratio of bits changed by the noise: 0%. Ratio of letters changed by the noise: 0%.
While the threefold repetition code corrects 1 error per codeword (of length 3), the Golay code corrects 3 errors per codewords (of length 24).
| Noisy encoded digital text | Error corrected noisy encoded text |
|---|---|
|
|