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Total anti-symmetrische Quasigruppen
Bei der Untersuchung von Prüfziffersystemen über Quasigruppen stößt man auf die so genannten total anti-symmetrischen Quasigruppen. Bislang war ihre Existenz für alle Ordnungen $4k+2\geq 10$ ungeklärt. Ecker und Poch vermuteten 1986, dass es keine total anti-symmetrischen Quasigruppen der Ordnung $4...
| Kaituhi matua: | |
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| Ētahi atu kaituhi: | |
| Hōputu: | Dissertation |
| Reo: | Tiamana |
| I whakaputaina: |
Philipps-Universität Marburg
2004
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| Ngā marau: | |
| Urunga tuihono: | Kuputuhi katoa PDF |
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Totally anti-symmetric quasigroups are employed in check digit systems. Until today their existence for all orders $4k+2\geq 10$ was unsettled. Ecker and Poch conjectured in 1986 that there are no totally anti-symmetric quasigroups of order $4k+2$. We disprove this conjecture and develop constructions for totally anti-symmetric quasigroups of order $n$ for all $n\neq 2,6$. By a computer search we prove in addition that check digit systems over a 2-quasigroup of the order 10, just as check digit systems over groups of order 10, cannot detect all (jump) twin errors or jump transpositions. As a further result we show that the class of totally anti-symmetric quasigroups is no variety.