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Keywords:
quasigroup; $n$-ary quasigroup; check character system; code; the system of the serial numbers of German banknotes
quasigroup; $n$-ary quasigroup; check character system; code; the system of the serial numbers of German banknotes
Summary:
It is well known that there exist some types of the most frequent errors made by human operators during transmission of data which it is possible to detect using a code with one check symbol. We prove that there does not exist an $n$-T-code that can detect all single, adjacent transposition, jump transposition, twin, jump twin and phonetic errors over an alphabet that contains 0 and 1. Systems that detect all single, adjacent transposition, jump transposition, twin, jump twin errors and almost all phonetic errors of the form $a0\rightarrow 1a$, $a\neq 0$, $a\neq 1$ over alphabets of different, and minimal size, are constructed. We study some connections between the properties of anti-commutativity and parastroph orthogonality of T-quasigroups. We also list possible errors of some types (jump transposition, twin error, jump twin error and phonetic error) that the system of the serial numbers of German banknotes cannot detect.
References:
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