Communicating Efficiently Coding Theory

Error Detection and Product Tagging

Product Tagging Product tags are used for easy identification of products as well as for tracking and inventory purposes.

Product Tagging Each time product tags are typed or scanned into computer or transmitted in some other fashion, there is a chance that one or more digits in the number will change or be rearranged as they move from one location to the other.

Product Tagging Most common types of transmission error:

Universal Product Code (UPC) This is a Universal Product Code (UPC). Nearly every item  that you purchase from a grocery store and department  store has a UPC on it somewhere. These “bar codes” were originally created to help grocery stores speed up the checkout process and keep better track of inventory.

Universal Product Code (UPC)

Universal Product Code (UPC) To avoid this error, we apply the method called CHECK DIGIT SCHEMES. An example is error-correcting codes, which are able to identify errors and then correct them automatically. Most check digit scheme append an extra digit or digits (check digit/s) to the product tag.

Universal Product Code (UPC) Manufacturer gets permission from the Universal Code Council (UCC) to join the UPC system and pays an annual fee. UCC issues the manufacturer identification number.

Universal Product Code (UPC) In a twelve-digit UPC, the first digit x1 identifies the type of product. The following table lists the different values of x1 and when each used. 𝑥 1 , 𝑥 2 ,…, 𝑥 12

Universal Product Code (UPC)

Universal Product Code (UPC) The second set of five digits identifies the manufacturer while the third set of five digits identifies the product. 𝑥 2 , 𝑥 3 , 𝑥 4 , 𝑥 5 , 𝑥 6 𝑥 7 , 𝑥 8 , 𝑥 9 , 𝑥 10 , 𝑥 11

Universal Product Code (UPC)

Universal Product Code (UPC) 𝑥 1 =0, 𝑥 2 =7, 𝑥 3 =7, 𝑥 4 =3, 𝑥 5 =3, 𝑥 6 =0 𝑥 7 =3, 𝑥 8 =4, 𝑥 9 =4, 𝑥 10 =5, 𝑥 11 =1, 𝑥 12 =7

Universal Product Code (UPC) The last digit x12 of the UPC is called a check digit. This digit lets the system determine if it scanned the number correctly or not. Given the eleven-digit string x1x2 … x11, the check digit x12 is appended such that the whole twelve-digit “codeword” satisfies 3 𝑥 1 + 𝑥 2 +3 𝑥 3 + 𝑥 4 +⋯+3 𝑥 11 + 𝑥 12 ≡0mod10

Modular Arithmetic The mathematical statement a ≡ b (mod n) (read as “a is congruent to b modulo n) means that the difference a – b is divisible by the integer n.

Modular Arithmetic AGAIN … a ≡ b (mod n) means a – b is divisible by the integer n.

Modular Arithmetic IT IS TRUE THAT … 61 ≡ 5 (mod 2) since 61 - 5 = 56 is divisible by the integer 2.

Modular Arithmetic IT IS TRUE THAT … 61 ≡ 7 (mod 9) since 61 - 7 = 54 is divisible by the integer 9.

Modular Arithmetic IS IT TRUE THAT … 75 ≡ 15 (mod 10) ?

Modular Arithmetic IS IT TRUE THAT … 75 ≡ 15 (mod 10) ? YES

Modular Arithmetic IS IT TRUE THAT … 78 ≡ 10 (mod 7) ?

Modular Arithmetic IS IT TRUE THAT … 78 ≡ 10 (mod 7) ? NO

Meaning of the Check Equation What does this statement mean? 3 𝑥 1 + 𝑥 2 +3 𝑥 3 + 𝑥 4 +⋯+3 𝑥 11 + 𝑥 12 ≡0mod10

Universal Product Code (UPC) 𝑥 1 =0, 𝑥 2 =7, 𝑥 3 =7, 𝑥 4 =3, 𝑥 5 =3, 𝑥 6 =0 𝑥 7 =3, 𝑥 8 =4, 𝑥 9 =4, 𝑥 10 =5, 𝑥 11 =1, 𝑥 12 =7

Universal Product Code (UPC) 𝑥 1 =0, 𝑥 2 =7, 𝑥 3 =7, 𝑥 4 =3, 𝑥 5 =3, 𝑥 6 =0 𝑥 7 =3, 𝑥 8 =4, 𝑥 9 =4, 𝑥 10 =5, 𝑥 11 =1, 𝑥 12 =7 3 𝑥 1 + 𝑥 2 +3 𝑥 3 + 𝑥 4 +⋯+3 𝑥 11 + 𝑥 12 ≡0mod10 3(0)+7+3(7)+3+3(3)+0+3(3)+ 4+3(4)+5+3(1)+7=80?≡0mod10

Universal Product Code (UPC) 3 𝑥 1 + 𝑥 2 +3 𝑥 3 + 𝑥 4 +⋯+3 𝑥 11 + 𝑥 12 ≡0mod10

Universal Product Code (UPC) 0 33878 20296 2 3 𝑥 1 + 𝑥 2 +3 𝑥 3 + 𝑥 4 +⋯+3 𝑥 11 + 𝑥 12 ≡0mod10

Universal Product Code (UPC) Is this a valid UPC? 3-70501-09110-4 3 𝑥 1 + 𝑥 2 +3 𝑥 3 + 𝑥 4 +⋯+3 𝑥 11 + 𝑥 12 ≡0mod10

Check Equation for UPC with 13 digits 𝑥 1 +3 𝑥 2 + 𝑥 3 +3 𝑥 4 +⋯+3 𝑥 12 + 𝑥 13 ≡0mod10

Check Equation for UPC with 13 digits 4+3(8)+0+3(4)+8+3(8)+9+3(0)+6+ 3(4)+2+3(1)+6=110≡0𝑚𝑜𝑑10

International Standard Book Number (ISBN) The international standard book number (ISBN) code is used throughout the world by publisher to identify properties of each book. The first nine digits represent information regarding its language, publisher and title. The 10th digit is the check digit. The whole 10 digits should satisfy 𝑥 1 +2 𝑥 2 +3 𝑥 3 +4 𝑥 4 +⋯+10 𝑥 10 ≡0mod11

International Standard Book Number (ISBN) 𝑥 1 +2 𝑥 2 +3 𝑥 3 +4 𝑥 4 +⋯+10 𝑥 10 ≡0mod11

International Standard Book Number (ISBN) Calculus, 9th Edition Thomas Finney ISBN 10: 0-201-53174-7 𝑥 1 +2 𝑥 2 +3 𝑥 3 +4 𝑥 4 +⋯+10 𝑥 10 ≡0mod11

International Standard Book Number (ISBN) Since January 1, 2007, ISBNs have contained thirteen digits. The ISBN-13 check digit must range from 0 to 9 and must be such that the sum of all the thirteen digits, each multiplied by its (integer) weight, alternating between 1 and 3, is a multiple of 10. That is, 𝑥 1 +3 𝑥 2 + 𝑥 3 +3 𝑥 4 +⋯+3 𝑥 12 + 𝑥 13 ≡0mod10

International Standard Book Number (ISBN) 𝑥 1 +3 𝑥 2 + 𝑥 3 +3 𝑥 4 +⋯+3 𝑥 12 + 𝑥 13 ≡0mod10

International Standard Book Number (ISBN) Calculus, 9th Edition Thomas Finney ISBN 13: 9 780201 531749 𝑥 1 +3 𝑥 2 + 𝑥 3 +3 𝑥 4 +⋯+3 𝑥 12 + 𝑥 13 ≡0mod10

International Standard Book Number (ISBN) In the ISBN 0-8218-2626-X 0 tells that the book is published in either the US, UK, Australia, NZ or Canada 8218 – identify the publishing company               code for American Mathematical Society 2626 -  the number given to each book X is the check digit

International Standard Book Number (ISBN) Suppose a book has number ISBN 0-387-94225-C. Find the value of the check digit C.

Quick Response Code (QR code) The Quick response (QR) code was in Japan By Denso-Wave in 1994 It is design to allow high speed component scanning. It is detected as a 2-dimensional digital image by a semi-conductor image sensor.

Quick Response Code (QR code) Unlike the one-dimensional UPC that is scanned by a narrow beam of light to extract data, the QR code is detected as a 2-dimensional digital image by a semi-conductor image sensor.

Quick Response Code (QR code) The sensor locates the three distinctive squares at the corners of the image, and uses a smaller square near the fourth corner to normalize the image for size, orientation, and angle of viewing. The small dots are then converted to binary numbers and their validity checked with an error-correcting code.

Quick Response Code (QR code) QR CODE GENERATOR https://www.qr-code-generator.com/

What’s NEXT?

Cryptography

PREPARE FOR THIS LESSON Read SECTION 3.2 Encryption Cryptosystems Shift Ciphers Substitution Ciphers Vigenere Ciphers Use the provided SLIDES to supplement your readings. Click “PLAY” to hear the voice over.

Your Teacher will provide you with a copy of the Cryptography Slides.

Study before attending the Master Class on February 14.