Summary of lecture 4 We are accustomed to count like 1,2,3,4,5,6,7,8,9,10,11,12,… This is called counting in base 10 – the second digit tells us how many lots of tens we have.

Summary of lecture 4 We are accustomed to count like 1,2,3,4,5,6,7,8,9,10,11,12,… This is called counting in base 10 – the second digit tells us how many lots of tens we have. We can count in any base we like – e.g., base 5 (tally charts); base 12 (dozens); base 60 (time); base 2 (binary).

Summary of lecture 4 We are accustomed to count like 1,2,3,4,5,6,7,8,9,10,11,12,… This is called counting in base 10 – the second digit tells us how many lots of tens we have. We can count in any base we like – e.g., base 5 (tally charts); base 12 (dozens); base 60 (time); base 2 (binary). Addition works in any base we like, e.g., 9 in base 10 = 1001 in base 2 6 in base 10 = 110 in base = 1111 in base 2 = 15 in base 10 = 9 + 6

MAT199: Math Alive Error corretcing and compression Ian Griffiths Mathematical Institute, University of Oxford, Department of Mathematics, Princeton University

MAT199: Math Alive Error corretcing and compression Ian Griffiths Mathematical Institute, University of Oxford, Department of Mathematics, Princeton University correcting

Error correction Check digits – U.S. Postal Service money order

Error correction Check digits – U.S. Postal Service money order This number is the remainder when we add the other ten numbers together and divide by 9

Error correction and encoding Check digits - ISBNs This number is solely to check for errors

Error correction and encoding Universal Product Codes (U.P.C.)

Error correction and encoding Universal Product Codes (U.P.C.) Other things that use check digits include: Bank routing numbers Airline tickets QR Codes

Binary encoding – data compression In data compression binary codes are assigned in a way that minimizes the number of digits required, by choosing short codes for the most frequently occurring data.

Binary encoding – data compression A compression algorithm converts data that is easy to use into a format that is easy to transport. A common compression algorithm is a zip file. When this is uncompressed the file is unzipped you recover the original information. In data compression binary codes are assigned in a way that minimizes the number of digits required, by choosing short codes for the most frequently occurring data.

Binary encoding – data compression A compression algorithm converts data that is easy to use into a format that is easy to transport. A common compression algorithm is a zip file. When this is uncompressed the file is unzipped you recover the original information. In some cases not all of the information is recovered from the encoding. In such cases the data is returned only approximately to its original state: In data compression binary codes are assigned in a way that minimizes the number of digits required, by choosing short codes for the most frequently occurring data.

Binary encoding – data compression A compression algorithm converts data that is easy to use into a format that is easy to transport. A common compression algorithm is a zip file. When this is uncompressed the file is unzipped you recover the original information. In some cases not all of the information is recovered from the encoding. In such cases the data is returned only approximately to its original state: e.g., Joint Photographic Expert Group (jpeg) files and Motion Picture Expert Group (mpeg) files. (Graphics interchange format (gif) files recover all original information upon decoding.) In data compression binary codes are assigned in a way that minimizes the number of digits required, by choosing short codes for the most frequently occurring data.

Binary encoding – CDs The audio information on a CD is encoded in ridges on the disc which are read by a laser.

Binary encoding – CDs The audio information on a CD is encoded in ridges on the disc which are read by a laser. The data is coded using a Reed-Soloman code. The laser may misread a bump so there need to be error-correcting codes to handle single-bit errors. To solve this problem, extra data bits are added that allow the drive to detect single-bit errors and correct these.

Binary encoding – CDs The audio information on a CD is encoded in ridges on the disc which are read by a laser. The data is coded using a Reed-Soloman code. The laser may misread a bump so there need to be error-correcting codes to handle single-bit errors. To solve this problem, extra data bits are added that allow the drive to detect single-bit errors and correct these. The laser must be able to move between songs, so data needs to be encoded into the music telling the drive where it is on the disc. This problem is solved using what is known as subcode data which encodes position on the disc and song title. A scratch on the CD might cause a whole packet of bytes to be misread (known as a burst error). We are able to recover from such an event by interleaving the data on the disc, so that it is stored non-sequentially around one of the disc's circuits.

We can represent the five-digit code pictorially: Here we have marked each codeword with a different box (oval, double oval, solid rectangle, dashed rectangle). The codewords and errors dont overlap, but dont cover the whole space either.

The Hamming code

Summary of lecture 5 We wish to correct for one-bit errors using the minimum amount of extra memory. We define a perfect code as one in which the total number codes achievable (either correct or one-bit corrupted) is equal to the total available number of code combinations.

The Hamming code The Hamming code is a perfect code.

Summary of lecture 6 The Reed–Solomon code offers sophisticated error correction used in CDs and transmission of data: Eight-bit strings are encoded into ten-bit codes (like the Hamming codes that encode 4-digit strings into 7-digit codes). Each word in the RS-code is itself 8-bits long. Interleaving divides up the data by splitting into five pieces.

The Voyager 1 space probe is now the most distant human-made object from Earth at almost 10 billion miles away. The Voyager 1 space probe

The Voyager 1 space probe is now the most distant human-made object from Earth at almost 10 billion miles away. The probe is still transmitting information. The signal reached is now one hundred quintillion times weaker than when it was only a mile away ( times weaker). The Voyager 1 space probe

The Voyager 1 space probe is now the most distant human-made object from Earth at almost 10 billion miles away. The probe is still transmitting information. The signal reached is now one hundred quintillion times weaker than when it was only a mile away ( times weaker). The total power used by the transmitter is less than one eighth of an electric kettle. The signal can still be decoded because of error-correction schemes. The Voyager 1 space probe

American Express Travelers Checks

Add together the first nine digits: = Divide this number by 9 and take the remainder: Subtract this number from 9: If there are no errors then this is your final digit – this is a check digit.

American Express Travelers Checks Add together the first nine digits: =30. Divide this number by 9 and take the remainder: 3. Subtract this number from 9: 9-3=6. If there are no errors then this is your final digit – this is a check digit.

Airline tickets

Divide the ten digit number (without the final check digit) by 7 and take the remainder:

Airline tickets Divide the ten digit number (without the final check digit) by 7 and take the remainder: = remainder. 7

Airline tickets Divide the ten digit number (without the final check digit) by 7 and take the remainder: = remainder. 7 The remainder is the check digit.

American banking system

For a routing number a 1 a 2 a 3 a 4 a 5 a 6 a 7 a 8 a 9 calculate: 7a 1 + 3a 2 + 9a 3 + 7a 4 + 3a 5 + 9a 6 + 7a 7 + 3a 8

American banking system For a routing number a 1 a 2 a 3 a 4 a 5 a 6 a 7 a 8 a 9 calculate: 7a 1 + 3a 2 + 9a 3 + 7a 4 + 3a 5 + 9a 6 + 7a 7 + 3a 8 Take the remainder when you divide by 10 and this is the check digit.

American banking system For a routing number a 1 a 2 a 3 a 4 a 5 a 6 a 7 a 8 a 9 calculate: 7a 1 + 3a 2 + 9a 3 + 7a 4 + 3a 5 + 9a 6 + 7a 7 + 3a 8 Take the remainder when you divide by 10 and this is the check digit. In example above, 7x1 + 3x2 + 9x1 + 7x3 + 3x0 + 9x1 + 7x5 + 3x7 = = 10 remainder 8 10

American banking system For a routing number a 1 a 2 a 3 a 4 a 5 a 6 a 7 a 8 a 9 calculate: 7a 1 + 3a 2 + 9a 3 + 7a 4 + 3a 5 + 9a 6 + 7a 7 + 3a 8 Take the remainder when you divide by 10 and this is the check digit. In example above, 7x1 + 3x2 + 9x1 + 7x3 + 3x0 + 9x1 + 7x5 + 3x7 = = 10 remainder 8 Check digit is 8. 10

The barcode encodes the numbers beneath it. In this example the barcode reads: separator left side: separator right side: separator Universal Product Codes (barcodes) Separator The binary code is then decoded using the following decryption:

Other uses for barcodes

ISBNs Check digit

ISBNs Check digit In ISBN codes we add the first number times 10, the second times 9, and so on: 10x3+9x8+8x8+7x0+6x5+5x3+4x1+3x0+2x1=217

ISBNs Check digit In ISBN codes we add the first number times 10, the second times 9, and so on: 10x3+9x8+8x8+7x0+6x5+5x3+4x1+3x0+2x1=217 We then divide this number by 11 and take the remainder 217/11=19 remainder 8.

ISBNs Check digit In ISBN codes we add the first number times 10, the second times 9, and so on: 10x3+9x8+8x8+7x0+6x5+5x3+4x1+3x0+2x1=217 We then divide this number by 11 and take the remainder 217/11=19 remainder 8. We then subtract this number from 11 to get our check digit, 3.

ISBNs What errors can we detect? We can detect ALL single errors. We can detect ALL transpositions.

ISBNs What errors can we detect? We can detect ALL single errors. We can detect ALL transpositions. The only inconvenience is if the check digit turns out to be 10 (double figures). In this case we write X for the check digit.

The ZIP bar code The ZIP bar code consists of 52 long and short bars at the bottom of a letter. The bars at the start and the end are just guard lines.

The ZIP bar code Every group of five digits encodes one digit using the following code:

The ZIP bar code Every group of five digits encodes one digit using the following code: This gives, in the previous example, the ZIP+4 code

The ZIP bar code Every group of five digits encodes one digit using the following code: This gives, in the previous example, the ZIP+4 code The 1 is a check digit. This makes sure the sum of all the digits is an exact multiple of 10. (In this case the sum of the digits is 40.)

The ZIP bar code There is also additional error correction: each group of five bars has exactly two long and three short bars.

The ZIP bar code There is also additional error correction: each group of five bars has exactly two long and three short bars. If you have an incorrect digit you can use the sum of the other digits to work out what this should be (by ensuring the sum of all digits is an exact multiple of 10).

The ZIP bar code There is also additional error correction: each group of five bars has exactly two long and three short bars. If you have an incorrect digit you can use the sum of the other digits to work out what this should be (by ensuring the sum of all digits is an exact multiple of 10). More recent codes use 12 digits rather than 10, though the sum still adds up to an exact multiple of 10.