A Permutation-Free Length 3 Decimal Check Digit Code

TL;DR

This paper introduces a novel length-3 decimal check digit code that is permutation-free, capable of detecting various errors, and allows full codeword recovery from any two digits, addressing limitations of earlier codes.

Contribution

It presents a new length-3 decimal check digit code that eliminates cyclic errors and enables codeword reconstruction from any two digits, improving upon Verhoeff's earlier designs.

Findings

Detects single, transposition, and adjacent twin errors

Allows full codeword recovery from any two digits

Eliminates cyclic and phonetic errors

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 codes 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. Apparently, not even a pair of such disjoint 3-digit codes is known. The code developed herein, has the property that the knowledge of any two digits is sufficient to determine the entire codeword even though their positions were unknown. 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.