Check digit scheme Verhoeff

Group Theory and the Verhoeff Check Digit Scheme... 

In 1969, J. Verhoeff [26] developed a check digit scheme that meets this goal and more. It not only catches all single-digit and transposition-of-adjacent-digits errors, but catches all of the error types listed in Table 1.2. It is also flexible, as it can be used with an identification number of any length. Information on the development and use of this scheme can be found in [3). [6], [7]. [10]. [II]. and [29). 

Definition 5.4.1. The Verhoeff Check Digit Scheme. Let aia2 -- On-iOn he an identification number with check digit a . The check digit a is appended to the number 0102 . such that the following equation is satisfied ... 

Suppose this error is not caught. Then applying the Verhoeff check digit scheme calculation to both the correct and the incorrect numbers will result in a true statement in each case. This results in... 

German Bundesbank Check Digit Scheme In 1990. the German Bundesbank (Federal Bank) began using a scheme based on the Verhoeff check digit scheme [6]. While it does use the familiar Cayley table for Dio (Table 5.11). the Bundesbank s variation has two differences ... 

One question still remains Why do many identification number systems still use check digit schemes that do not catch all of the errors listed in Table 1.2 This is a good question for which I have no answer. However, it does motivate the need to adjust current schemes and to create new ones. All of the concepts introduced in this book can be applied to create sophisticated schemes that will catch most errors. This was demonstrated by the Verhoeff scheme. With the birth of new identification number systems and the existence of old and ineffective check digit schemes, innovative schemes must be developed to ensure that data and information continue to flow error-free. 

The discussion in Chapter 4 is focused on symmetry and rigid motions. The notation e.stabli.shed in Chapter 3 for permutations is used in a mathematical investigation of the symmetries of a variety of different shapes. The symmetries of a pentagon form the basts of the very reliable Verhoeff check digit scheme presented in Chapter 5. Furthermore, the use of rigid motions to create elaborate patterns will serve as an introduction to the discussion of group theory that begins Chapter 5. 

In check digit calculations for the other schemes studied, digits were multiplied or added together For example, 3-4 = 12or3 + 4 = 7 would be calculated. In the Verhoeff scheme, digits are combined using the results of which are presented in the Cayley table (Table 5.10) for Dm- For example 3 4 = 2 and 8 3 = 5. 

Example 5.4.4. Consider the five-digit number 27163 that has been generated to identify a certain item. The Verhoeff scheme will be used to append a check digit K to the number 27163 to create the identification number 27163/f. 

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