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
Available: Pr-A-98-17.ps
Contact: Schulz, Ralph-Hardo, Freie Universität Berlin, Fachbereich Mathematik und Informatik, Arnimallee 2-6, D-14195 Berlin, Germany (schulz@math.fu-berlin.de)
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