Check character systems

Abstract

--ZusammenfassungCheck Character Systems. A hopefully complete review is given of different decimal codes that are in use. The grouptheoretical background of those methods is explained and applied to the construction of new error detecting codes. With the help of Latin squares (or quasi-groups) new error codes are found, especially a good decimal shift-code.Priifzeiehen-Verfahren. Wir geben einen m6glichst vollstfindigen Oberblick fiber alle gebrfiuchlichen Priifziffern-Verfahren. Der gruppentheoretische Hintergrund aller dieser Verfahren wird erklgrt, und es wird gezeigt, wie sich auf dieser Basis neue Prtifzeichen-Verfahren konstruieren lassen. Mit Hilfe lateinischer Quadrate (oder Quasigruppen) werden neue Priifzeichen-Verfahren gefunden, speziell ein guter dezimaler Shift-Code.

Optimal Check Digit Systems Based on Modular Arithmetic

“…They are the most widely disseminated [1], [5], [6], [7] and [11]. In these systems, arithmetic operations are conducted in the digits of identifiers and the results are considered as mod n, for n ∈ N. Given an identifier with m digits a 1 a 2 a 3 · · · a m and a sequence of weights p i , the check digit a m is determined by the direct system equation given by: In a more comprehensive manner, we shall, rather than attributing weights to check digits, define functions σ i , for each position 1 ≤ i ≤ m − 1.…”

“…Until 2013 there have been several attempts to find such a system, all of them are described in [5], but none of them has achieved the optimal method.…”

Check Character Systems using Quasigroups: I

“…In this work decimal codes known in the 1970s are presented. Ecker and Poch in [6] have given a survey of check character systems and their analysis from a mathematical point of view. In particular, the group-theoretical background of the known methods was explained and new codes were presented that stem from the theory of quasigroups.…”

“…In [6] for this code necessary and sufficient conditions are established to detect a transposition and a jump transposition (but only in the information symbols). In order to increase the possibilities of locating suitable codes by means of a quasigroup Q(·) in [6] it is suggested to complicate check formula (1) in the following way: a n = (.…”

“…In Section 5 of [6] the possibility of constructing error detecting codes based on Latin squares (or on quasigroups) is investigated. The error detecting capability of such code may (to some extent) be better than of a modulo m check system.…”