DOI QR코드

DOI QR Code

LINEAR AND NON-LINEAR LOOP-TRANSVERSAL CODES IN ERROR-CORRECTION AND GRAPH DOMINATION

  • Dagli, Mehmet (Department of Mathematics Amasya University) ;
  • Im, Bokhee (Department of Mathematics Chonnam National University) ;
  • Smith, Jonathan D.H. (Department of Mathematics Iowa State University)
  • Received : 2019.02.21
  • Accepted : 2019.08.14
  • Published : 2020.03.31

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.

Acknowledgement

Supported by : National Research Foundation of Korea (NRF)

References

  1. R. Aydinyan, Loop Transversal Codes over Finite Rings, ProQuest LLC, Ann Arbor, MI, 2005.
  2. R. Aydinyan and J. D. H. Smith, Loop transversal codes for error detection, J. Combin. Math. Combin. Comput. 58 (2006), 153-159.
  3. D.-H. Choi and J. D. H. Smith, Greedy loop transversal codes for correcting error bursts, Discrete Math. 264 (2003), no. 1-3, 37-43. https://doi.org/10.1016/S0012-365X(02)00548-4
  4. F.-L. Hsu, F. A. Hummer, and J. D. H. Smith, Logarithms, syndrome functions, and the information rates of greedy loop transversal codes, J. Combin. Math. Combin. Comput. 22 (1996), 33-49.
  5. F. A. Hummer, Loop Transversal Codes, ProQuest LLC, Ann Arbor, MI, 1992.
  6. F. A. Hummer and J. D. H. Smith, Greedy loop transversal codes, matrices, and lexi-codes, J. Combin. Math. Combin. Comput. 22 (1996), 143-155.
  7. I. Marquez-Corbella, A combinatorial commutative algebra approach to complete decoding, Ph.D. Thesis, Universidad de Valladolid, 2013.
  8. N. Obradovic, J. Peters, and G. Ruzic, Ecient domination in circulant graphs with two chord lengths, Inform. Process. Lett. 102 (2007), no. 6, 253-258. https://doi.org/10.1016/j.ipl.2007.02.004
  9. J. D. H. Smith, Loop transversals to linear codes, J. Combin. Inform. System Sci. 17 (1992), no. 1-2, 1-8.
  10. J. D. H. Smith and A. B. Romanowska, Post-Modern Algebra, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1999. https://doi.org/10. 1002/9781118032589