Optimal Check Digit Systems Based on Modular Arithmetic

Luerbio Faria, Paulo Eustáquio Duarte Pinto, Natália Pedroza

Abstract


In this article, we discuss check digits systems based on modular arithmetic, which are used worldwide. Check digits were created to eliminate most errors in data input of computational systems. Though old, a discussion about the optimality of the systems used is not found in the literature. We describe the main existing systems worldwide and highlight those adopted in Brazil. We present the necessary improvements in order to make all systems optimal. We also propose a new optimal system with 3 permutations for systems with modular base 10.

Keywords


Check digits; Error detection; Modular arithmetic

Full Text:

PDF

References


D. Argenta, and R. Amorim. Estudo e Implementação de Dígitos Verifi-

cadores. 2012. 52 f. Monograph (Computer Science). Universidade do Estado do Rio de Janeiro, 2012.

G. B. Belyavskaya, V. I. Isbash and G. L. Mullen. Check Character Systems Using Quasigroups: I. Des. Codes Cryptography, v. 37, n.2, p. 215-227, 2004.

H. M. Damm. Check digit systems over groups and anti-symmetric mappings. Archiv Der Mathematik, v. 75, n.6, p. 413-421, 2000.

H. M. Damm. Totally anti-symmetric quasigroups for all orders n 6= 2, 6.

Discrete mathematics, v. 307, n. 6, p. 715–729 , 2007.

A. Ecker and G. Poch. Check Character Systems. Computing, v. 37, n. 4, p. 277–301, 1984.

J. A. Gallian Error detection methods. ACM Computing Surveys (CSUR), v. 28, n. 3, p. 504-517, 1996.

J. A. Gallian Contemporary Abstract Algebra, 70. ed., Cengage Learning, 2010.

P. Gumm. A new class of check digit methods for arbitrary number systems. Information Theory, IEEE Transactions on v. 31, n. 1, p. 102-105, 1985.

H. P. Luhn. Computer for verifying numbers. US n. 2.950.048, 6 January 1954, 23 August 1960.

IBAN, Available in http://www.tbg5-finance.org/?ibancheck.shtml. Last accessed 10 october 2016.

M. Malek. Coding Theory, Decimal Code. Available in

http://www.mcs.csueastbay.edu/ malek/TeX/Decimal.pdf, 2009. Last

accessed 10 october 2016.

N. Pedroza. Uma análise dos esquemas de dígitos verificadores usados no Brasil. 84f. Master’s thesis. Universidade do Estado do Rio de Janeiro, 2013.

R.-H. Schulz. On Check Digit Systems using Antisymmetric Mappings. Mathematik und Informatik, Freie Universät Berlin, 1999.

R.-H. Schulz. Check Character Systems and Anti-symmetric Mappings. Computational Discrete Mathematics, Advanced Lectures, p. 136-147, 2001.

J. Verhoeff. Error Detecting Decimal Codes. 122 f. Tese, Mathematical Centre Tract 29, The Mathematical Centre, 1969.

E. F. Wood. Self-Checking Codes - An Application of Modular Arithmetic. The Mathematics Teacher, v. 80, n. 4, p. 312-316, 1987.




DOI: http://dx.doi.org/10.5540/tema.2017.018.01.0105

Article Metrics

Metrics Loading ...

Metrics powered by PLOS ALM

Refbacks

  • There are currently no refbacks.



TEMA - Trends in Applied and Computational Mathematics

A publication of the Brazilian Society of Applied and Computational Mathematics (SBMAC)
ISSN: 1677-1966  (print version),  2179-8451  (online version)

Indexed in:

                        

 

Desenvolvido por:

Logomarca da Lepidus Tecnologia