## Gary L. Mullen, Victor Shcherbacov

*$n$-T-quasigroup codes with one check symbol and their error detection capabilities *

Comment.Math.Univ.Carolinae 45,2 (2004) 321-340. **Abstract:**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 $a0\rightarrow 1a$, $a\not =0$, $a\not =1$ 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.

**Keywords:** quasigroup, $n$-ary quasigroup, check character system, code, the system of the serial numbers of German banknotes

**AMS Subject Classification:** 94B60, 94B65, 20N05, 20N15

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