On check digit systems using anti-symmetric mappings. (English) Zbl 0967.94030

Schulz, Ralph-Hardo

Althöfer, Ingo (ed.) et al., Numbers, information and complexity. Dedicated to Rudolf Ahlswede on the occasion of his 60th birthday. Dordrecht: Kluwer Academic Publishers. 295-310 (2000).

This paper is a survey on check digit systems over a group having check equation \(T(a_1) T(a_2)\dots\)
\(T(a_n)= e\), where \(a_1, a_2\dots a_n\) is a codeword consisting of group elements, \(e\) is a fixed element of the group (in most cases the neutral element), and \(T\) a permutation acting on the group. The paper starts with an overview of commonly made errors in sequences of digits, from which it appears that single errors and adjacent transposition are the most important ones. In order to detect such errors the permutation \(T\) should be antisymmetric i.e. \(xT(y)\neq yT(x)\) for all \(x, y\) in the group. An extensive list of results on groups allowing for antisymmetric mappings follows. Moreover equivalence relations between check digit systems are discussed and described in detail for the case of the group \(D_5\). Finally the performance on the detection of other types of errors is touched upon.
For the entire collection see [Zbl 0933.00056].

Reviewer: H.J.Tiersma (Diemen)

MSC:

94B60

Other types of codes

Keywords:

error detection; survey; check digit systems; antisymmetric mappings