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

Gary L. Mullen; Viktor Alekseevich Shcherbakov

Commentationes Mathematicae Universitatis Carolinae (2004)

- Volume: 45, Issue: 2, page 321-340
- ISSN: 0010-2628

## Access Full Article

top## Abstract

top## How to cite

topMullen, Gary L., and Shcherbakov, Viktor Alekseevich. "$n$-T-quasigroup codes with one check symbol and their error detection capabilities." Commentationes Mathematicae Universitatis Carolinae 45.2 (2004): 321-340. <http://eudml.org/doc/249383>.

@article{Mullen2004,

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 $a0\rightarrow 1a$, $a\ne 0$, $a\ne 1$ 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.},

author = {Mullen, Gary L., Shcherbakov, Viktor Alekseevich},

journal = {Commentationes Mathematicae Universitatis Carolinae},

keywords = {quasigroup; $n$-ary quasigroup; check character system; code; the system of the serial numbers of German banknotes; -ary quasigroup; check character system; code; phonetic error},

language = {eng},

number = {2},

pages = {321-340},

publisher = {Charles University in Prague, Faculty of Mathematics and Physics},

title = {$n$-T-quasigroup codes with one check symbol and their error detection capabilities},

url = {http://eudml.org/doc/249383},

volume = {45},

year = {2004},

}

TY - JOUR

AU - Mullen, Gary L.

AU - Shcherbakov, Viktor Alekseevich

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

JO - Commentationes Mathematicae Universitatis Carolinae

PY - 2004

PB - Charles University in Prague, Faculty of Mathematics and Physics

VL - 45

IS - 2

SP - 321

EP - 340

AB - 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 $a0\rightarrow 1a$, $a\ne 0$, $a\ne 1$ 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.

LA - eng

KW - quasigroup; $n$-ary quasigroup; check character system; code; the system of the serial numbers of German banknotes; -ary quasigroup; check character system; code; phonetic error

UR - http://eudml.org/doc/249383

ER -

## References

top- Beckley D.F., An optimum system with modulo $11$, The Computer Bulletin 11 213-215 (1967). (1967)
- Belousov V.D., Foundations of the Theory of Quasigroups and Loops, Nauka, Moscow, 1967 (in Russian). MR0218483
- Belousov V.D., Elements of the Quasigroup Theory, A Special Course, Kishinev, 1981 (in Russian).
- Belousov V.D., $n$-Ary Quasigroups, Shtiinta, Kishinev, 1972 (in Russian). MR0354919
- Belyavskaya G.B., Izbash V.I., Mullen G.L., Check character systems using quasigroups, I and II, preprints. MR2174275
- Damm M., Prüfziffersysteme über Quasigruppen, Diplomarbeit, Philipps-Universität Marburg, 1998.
- Dénes J., Keedwell A.D., Latin Squares and their Applications, Académiai Kiadó, Budapest, 1974. MR0351850
- Ecker A., Poch G., Check character systems, Computing 37/4 277-301 (1986). (1986) Zbl0595.94012MR0869726
- Gumm H.P., A new class of check-digit methods for arbitrary number systems, IEEE Trans. Inf. Th. IT, 31 (1985), 102-105. Zbl0557.94013
- Kargapolov M.I., Merzlyakov Yu.I., Foundations of Group Theory, Nauka, Moscow, 1977 (in Russian). Zbl0508.20001MR0444748
- Laywine Ch.L., Mullen G.L., Discrete Mathematics using Latin Squares, John Wiley & Sons, Inc., New York, 1998. Zbl0957.05002MR1644242
- Mullen G.L., Shcherbacov V., Properties of codes with one check symbol from a quasigroup point of view, Bul. Acad. Ştiinte Repub. Mold. Mat. 2002, no 3, pp.71-86. Zbl1065.94021MR1991018
- Pflugfelder H.O., Quasigroups and Loops: Introduction, Heldermann Verlag, Berlin, 1990. Zbl0715.20043MR1125767
- Sade A., Produit direct-singulier de quasigroupes othogonaux et anti-abeliens, Ann. Soc. Sci. Bruxelles, Ser. I, 74 (1960), 91-99. (1960) MR0140599
- Schulz R.-H., Check Character Systems and Anti-symmetric Mappings, H. Alt (Ed.): Computational Discrete Mathematics, Lecture Notes in Comput. Sci. 2122, 2001, pp.136-147. Zbl1003.94537MR1911586
- Schulz R.-H., Equivalence of check digit systems over the dicyclic groups of order $8$ and $12$, in J. Blankenagel & W. Spiegel, editor, Mathematikdidaktik aus Begeisterung für die Mathematik, pp.227-237, Klett Verlag, Stuttgart, 2000. Zbl1011.94539
- Verhoeff J., Error Detecting Decimal Codes, Vol. 29, Math. Centre Tracts. Math. Centrum Amsterdam, 1969. Zbl0267.94016MR0256770

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.