Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

Article

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Bull. Korean Math. Soc. 2020; 57(2): 295-309

Online first article January 7, 2020      Printed March 31, 2020

https://doi.org/10.4134/BKMS.b190204

Copyright © The Korean Mathematical Society.

Linear and non-linear loop-transversal codes in error-correction and graph domination

Mehmet Da\v{g}l{\i}, Bokhee Im, Jonathan D. H. Smith

Amasya University; Chonnam National University; Iowa State University

Abstract

Loop transversal codes take an alternative approach to the theory of error-correcting codes, placing emphasis on the set of errors that are to be corrected. Hitherto, the loop transversal code method has been restricted to linear codes. The goal of the current paper is to extend the conceptual framework of loop transversal codes to admit nonlinear codes. We present a natural example of this nonlinearity among perfect single-error correcting codes that exhibit efficient domination in a circulant graph, and contrast it with linear codes in a similar context.

Keywords: Loop transversal code, nonlinear code, quasigroup, loop, circulant graph, efficient domination

MSC numbers: 94B60, 20N05, 05C69

Supported by: The second author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF), funded by the Ministry of Education (NRF-2017R1D1A3B05029924).