Totally Disjoint 3-Digit Decimal Check Digit Codes
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.
Cite
@article{arxiv.2504.05326,
title = {Totally Disjoint 3-Digit Decimal Check Digit Codes},
author = {Larry A. Dunning},
journal= {arXiv preprint arXiv:2504.05326},
year = {2025}
}