Chapter 16: Check Digit Systems, Continued

1. MAT 105 Spring 2008 Chapter 16: Check Digit Systems, Continued

2. A New Method • The methods we’ve been using so far are not great at detecting transposition errors • Since these errors are relatively common, we want to find a system that can detect all possible transpositions • The method we will use involves a weighted sum

3. UPC (Universal Product Code) Here is an example of a UPC from a typical product Notice that there are 12 digits: a single digit, two groups of 5, and another single digit check digit category of goods product ID manufacturer ID

4. Parts of the UPC • The first number represents the general “category of goods” • Most fixed-weight products are in category 0 • Coupons are in category 5 • The next 5 digits identify the manufacturer • For example, Coca-Cola is 49000 • The next 5 digits identify the particular product • For example, a 12 oz. can of Diet Coke is 01134 • The last digit is the check digit

5. Computing the Check Digit • Instead of adding all of the digits together, we do something a little more complex • Multiply the first digit by 3 • Add the second digit • Multiply the third digit by 3 • Add the fourth digit • etc. • The check digit is chosen so that this sum ends in 0 • Notice that we include the check digit in our sum!

6. Weighted Sums • This is called a weighted sum • In a weighted sum, we multiply the digits by “weights” before adding them together • In this case, the weights are: • 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1

7. Examples • Show that 0-58200-48826-7 is valid • Show that 0-52200-48826-7 is invalid • The UPC system detects all substitution errors and 89% of all other kinds of errors

8. Another System: Bank Routing Numbers

9. What’s a Routing Number? • A routing number uniquely identifies a bank • If you have direct deposit on your paychecks, your employer asks you for your routing number (or for a cancelled check to read the number from) • Also listed on your checks are your account number and the specific check number

10. Check Digits on Checks • Routing numbers are 8 digits long • The 9th digit is a check digit • The check digit is the last digit of the weighted sum of the first 8 digits with weights 7, 3, 9, 7, 3, 9, 7, 3

11. Examples • Show that 111000025 is a valid routing number (this is the routing number for a bank in Texas) • Our weighted sum is 7*1 + 3*1 + 9*1 + 7*0 + 3*0 + 9*0 + 7*0 + 3*2 = 25 • The last digit of the weighted sum is 5, so the routing number is valid!

12. Another Example • Show that 231381116 is a valid routing number (this is the routing number for PSECU) • Show that 238311116 is not valid • What kind of error was committed?

13. Why Are More Complex Systems Better? • The bank routing number check digit system is fairly complex, but it is better than the UPC system • The UPC system cannot detect jump transpositions at all, but the routing number system can • The more errors a system can detect, the better • More complex systems can detect more errors

14. Codabar • The system used by credit cards is called Codabar • A credit card number is 16 digits long • The first 15 digits identify the credit card, and the 16th digit is the check digit

15. The Codabar Process • Add the digits in the odd-numbered positions (1st, 3rd, 5th, etc.) • Double this sum • Add to this total the number of odd-position digits that are above 4 (add the number of digits, not the digits themselves) • Add the remaining (even position) digits • The check digit is chosen so that the total ends in 0

16. An Example • Consider the credit card number4128 0012 3456 7890 • Let’s check to make sure this credit card number is valid using Codabar

17. Running Through the Process • Add the digits in the odd-numbered positions • Double this sum • Add to this total the number of odd-position digits that are above 4 • Add the remaining (even position) digits • The check digit is chosen so that the total ends in 0 • 4128 0012 3456 78964+2+0+1+3+5+7+9 = 31 • 31 x 2 = 62 • 4128 0012 3456 789662 + 3 = 65 • 4128 0012 3456 789665+1+8+0+2+4+6+8+6 = 100 

18. Finding the Check Digit • What if the company is trying to determine which check digit should be appended to a given ID number? • 3125 6001 9643 001_ • What should the check digit be?

19. Benefits of Codabar • The Codabar method detects all substitution errors and 98% of all other common errors • This is important since credit card numbers are one of the more universal ID numbers we use on a daily basis

20. ISBN: International Standard Book Number • An ISBN is a 10-digit number • For example, the ISBN for our textbook is 0 – 7167 – 5965 – 9 check digit Indicates that the book was published in an English-speaking country publisher ID book ID

21. Weighted Sums Again • For an ISBN number, we compute a weighted sum with weights 10, 9, 8, 7, 6, 5, 4, 3, 2, 1 • The check digit is chosen so that the sum is evenly divisible by 11

22. An Example • Let’s check the ISBN 0-7167-5965-9 • We compute our weighted sum to be 253, which is evenly divisible by 11 • Which check digit would you need for ISBN0-7167-1910-_ ?

23. An 11th Digit • Since we’re using division by 11, sometimes we’ll need an 11th digit • When the check digit would need to be 10, we use the letter X instead

24. Benefits of ISBN • The ISBN system detects all substitution errors and all transposition errors • Since valid ISBN’s were running out, new books are published using a new 13-digit system with a different way of computing the check digit • You might explore these ideas in the third writing assignment