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arXiv · 2504.05326

Totally Disjoint 3-Digit Decimal Check Digit Codes

Larry A. Dunning
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Abstract

In 1969 J. Verhoeff provided the first examples of a decimal error detecting code using a single check digit to provide protection against all single, transposition and adjacent twin errors. The three versions of such a code that he presented are length 3-digit codes with 2 information digits. Existence of a 4-digit code would imply the existence of 10 such disjoint 3-digit codes. This paper presents 3 pairwise disjoint 3-digit codes. The codes developed herein, have the property that the knowledge of the multiset of digits included in a word is sufficient to determine the entire codeword even though their positions were unknown. Thus the codes are permutation-free, and this fulfills Verhoeff's desire to eliminate "cyclic errors". Phonetic errors, where 2 digit pairs of the forms X0 and 1X are interchanged, are also eliminated.

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Larry A. Dunning. 2025-07-31. Totally Disjoint 3-Digit Decimal Check Digit Codes. https://arxiv.org/abs/2504.05326

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