## Example: The Verhoeff check digit scheme

We have seen in §3.3.2 the ISBN check digit scheme. This helps detect an error made in one of the digits. In 1969, J. Verhoeff [V] describes a check digit scheme using the dihedral group.

Let . This group has elements, each element corresponding to the digits . For example, if are the elements of in some order (it will be typographically simpler, as you will see, to use instead of for our notation) then we can associate to . We let so for example .

Verhoeff check digit scheme ([Ki], §5.4) Fix an integer and fix an element in the symmetric group of the set . Let be a identification number without a check digit. The digit is appended to provided (Actually, Verhoeff uses decending powers of in his scheme. We use ascending powers since it is easier to remember and because the German Bundesbank used this version for the Deutsche Mark - see [Ki] for more details.)

Example 5.10.3   Let and let .

Consider the ID number , what is the check digit? We compute , , , , . Thus is determined by Since , we have . The ID number with check digit is .

Consider the ID number , what is the check digit? Answer: . The ID number with check digit is . We leave it as an exercise to check this (see Exercise 5.10.4).

David Joyner 2007-09-03