Preprint No. A-99-10

Ralph-Hardo Schulz

Equivalence of check digit systems over the dicyclic groups of order 8 and 12

Abstract: We consider check digit systems $(G,T)$ over a group $G$ using check equation $\Prod^{n}_{i=1}T^{\,i}(c_{i})=1$ for codewords $c_{1}\ldots c_{n}\in G^{\,n}$ and an anti--symmetric mapping $T$, i.e. a permutation $T$ of $G$ with $xT(y)\neq yT(x)$ for all $x,y\in G$, $x\neq y$. Such a system allows to detect all single errors (i.e. errors in only one component) and neighbour transpositions (i.e. errors made by interchanging adjacent components). \noindent Two such systems $(G,T_{1})$ and $(G,T_{2})$ are called automorphism equivalent if $T_{2}=\alpha^{-1}\circ T_{1}\circ\alpha$ for an automorphism $\alpha$ of $G$. In particular, we describe anti--symmetric mappings, automorphism equivalence classes and their error detection rates for three special groups, the dihedral group $D_{5}$ (of order 10), the quaternion group $Q_{2}$ (of order 8) and the dicyclic group $Q_{3}$ (of order 12); these results are part of a cooperation with Sabine Giese and Sehpahnur Ugan.

Keywords: check digit systems, check chracter systems, error detecting codes, anti-symmetric mappings, automorphisms, equivalence of codes, dihedral groups, dicyclic groups, quaternion group

Mathematics Subject Classification (MSC91): 94B60 Other types of codes

Language: ENG

Available: Pr-A-99-10.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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