Preprint No. A-98-17

Ralph-Hardo Schulz

On Check Digit Systems using Anti-symmetric Mappings

Abstract: We consider check digit systems over a group $G$ with check equation $T(a_{1}) T^{2}(a_{2}) \cdots T^{n}(a_{n})=e$ for fixed $e \in G$ and permutation $T$ of $G$. Such a system detects all single errors (i.e. errors in only one component); and it detects adjacent transpositions (i.e. errors of the form $ \ldots ab \ldots \longrightarrow \ldots ba \ldots$) iff $T$ %$= \delta_{i+1}\delta_{i}^{-1}$ is anti--symmetric that means that $T$ fulfills the condition $ x \mbox{ T}(y)\neq y \mbox{ T}(x) \mbox{~~for all~~ }x, y \in G \mbox{~~with~} x \neq y .$ In this survey we shall report on the existence of groups with anti--symmetric mappings, define equivalence relations between check digit systems and describe, in the special case of the dihedral group $D_{5}$, the equivalence classes.

Keywords: check digit system, check character system, error detecting code, anti--symmetric mapping, complete mapping, orthomorphism, dihedral group, Chevalley group.

Mathematics Subject Classification (MSC91): 94B99 None of the above but in this section , 20B40 Computational methods

Language: ENG


Contact: Schulz, Ralph-Hardo, Freie Universität Berlin, Fachbereich Mathematik und Informatik, Arnimallee 2-6, D-14195 Berlin, Germany (

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