Verhoeff algorithm

The Verhoeff algorithm is a checksum formula for error detection developed by the Dutch mathematician Jacobus Verhoeff and was first published in 1969. It was the first decimal check digit algorithm which detects all single-digit errors, and all transposition errors involving two adjacent digits, which was at the time thought impossible with such a code.

Verhoeff had the goal of finding a decimal code—one where the check digit is a single decimal digit—which detected all single-digit errors and all transpositions of adjacent digits. At the time, supposed proofs of the nonexistence of these codes made base-11 codes popular, for example in the ISBN check digit.

His goals were also practical, and he based the evaluation of different codes on live data from the Dutch postal system, using a weighted points system for different kinds of error. The analysis broke the errors down into a number of categories: first, by how many digits are in error; for those with two digits in error, there are transpositions (ab → ba), twins (aa → 'bb'), jump transpositions (abc → cba), phonetic (1a → a0), and jump twins (aba → cbc). Additionally there are omitted and added digits. Although the frequencies of some of these kinds of errors might be small, some codes might be immune to them in addition to the primary goals of detecting all singles and transpositions.

The phonetic errors in particular showed linguistic effects, because in Dutch, numbers are typically read in pairs; and also while 50 sounds similar to 15 in Dutch, 80 doesn't sound like 18.

Taking six-digit numbers as an example, Verhoeff reported the following classification of the errors:.

Verhoeff devised his algorithm using the properties of the dihedral group of order 10 (a non-commutative system of operations on ten elements, which corresponds to the rotation and reflection of a regular pentagon), combined with a permutation. He claimed that it was the first practical use of the dihedral group, and confirmed the principle that in the end, all beautiful mathematics will find a use, even though in practice the algorithm will be implemented using simple lookup tables without needing to understand how to generate those tables from the underlying group and permutation theory.

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