# Error Detection Based on Check Digit Schemes

**Jeffrey Walton**on

**December 13th, 2006**

### Introduction

A Check Digits is an alphanumeric character added to a number to detect human errors. Check Digits are a simple and easy way to neutralize the human element of keying data. Check Digits are not Error Correcting Codes. Should the reader want to investigate robust Error Correcting Codes, he or she is referred to *Error-Correcting Codes* by Peterson and Weldon.

It is also notable that CRC codes are not a check digit scheme because more than one alphanumeric digit is generated to detect errors.

If the reader desires a system for very long datasets or binary data sets, consider using a CRC, Alder checksums, or a hash-based checksum.

According to Richard Hamming, the two most common errors are:

- 12 becomes 21 (adjacent characters are transposed)
- 112 becomes 122 (incorrect doubling of triples)

Jacobus Verhoeff presents the follow statistics in *Error Detecting Decimal Codes*:

Error | Approximate Percentage | Comment |
---|---|---|

a → b | 60% - 90% | Single Error |

ab → aab | 10% - 20% | Adding a Digit |

aab → ab | 10% - 20% | Omitting a Digit |

ab → ba | 10% - 20% | Transposition Error |

aa → bb | 0.5% - 1.5% | Twin Errors |

acb → bca | > 1% | Jump Twin Error |

13 → 30 | 0.5% - 1.5% | Phonetic Error (similar pronunciations) |

This article discusses the following Check Digit Schemes:

- Mod 9 Scheme
- Mod 7 Scheme
- Mod 11 Scheme
- 3-Weight Method
- IBM Scheme
- UPC Scheme
- Verhoeff Algorithm
- ISO 7064 Mod N/N+1