Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-26T21:09:48.476Z Has data issue: false hasContentIssue false
  • English
  • Français

ELUSIVE CODES IN HAMMING GRAPHS

Part of: Theory of error-correcting codes and error-detecting codes Algebraic combinatorics

Published online by Cambridge University Press:  28 February 2013

DANIEL R. HAWTIN ,
NEIL I. GILLESPIE  and
CHERYL E. PRAEGER
DANIEL R. HAWTIN*
Affiliation:
The University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia email [email protected]@uwa.edu.au
NEIL I. GILLESPIE
Affiliation:
The University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia email [email protected]@uwa.edu.au
CHERYL E. PRAEGER
Affiliation:
The University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia email [email protected]@uwa.edu.au Also affiliated with King Abdulaziz University, Jeddah, Saudi Arabia email [email protected]
*
✉[email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider a code to be a subset of the vertex set of a Hamming graph. We examine elusive pairs, code-group pairs where the code is not determined by knowledge of its set of neighbours. We construct a new infinite family of elusive pairs, where the group in question acts transitively on the set of neighbours of the code. In these examples, the alphabet size always divides the length of the code. We show that there is no elusive pair for the smallest set of parameters that does not satisfy this condition. We also pose several questions regarding elusive pairs.

Keywords

elusive codespermutation codespowerline communicationneighbour transitive codesautomorphism groups

MSC classification

Secondary: 94B60: Other types of codes 05E18: Group actions on combinatorial structures
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 88 , Issue 2 , October 2013 , pp. 286 - 296
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

References

Bailey, R. F., ‘Error-correcting codes from permutation groups’, Discrete Math. 309 (2009), 4253–4265.CrossRefGoogle Scholar
Blake, I. F., ‘Permutation codes for discrete channels’, IEEE Trans. Inform. Theory 20 (1974), 138–140.CrossRefGoogle Scholar
Blake, I. F., Cohen, G. and Deza, M., ‘Coding with permutations’, Inf. Control 43 (1979), 1–19.CrossRefGoogle Scholar
Brouwer, A. E., Cohen, A. M. and Neumaier, A., Distance-Regular Graphs, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 18 (Springer, Berlin, 1989).CrossRefGoogle Scholar
Chu, W., Colbourn, C. J. and Dukes, P., ‘Constructions for permutation codes in powerline communications’, Des. Codes Cryptogr. 32 (1–3) (May 2004), 51–64.CrossRefGoogle Scholar
Gillespie, N. I., Neighbour Transitivity on Codes in Hamming Graphs, PhD Thesis, The University of Western Australia, Perth, Australia, 2011.Google Scholar
Gillespie, N. I. and Praeger, C. E., ‘From neighbour transitive codes to frequency permutation arrays’. 2012. arXiv:1204.2900v1.Google Scholar
Gillespie, N. I. and Praeger, C. E., ‘Neighbour transitivity on codes in Hamming graphs’, Des. Codes Cryptogr., published online February 2012. doi:10.1007/s10623-012-9614-5.CrossRefGoogle Scholar
Huczynska, S., ‘Powerline communication and the 36 officers problem’, Phil. Trans. R. Soc. A 364 (2006), 3199–3214.CrossRefGoogle ScholarPubMed
Huffman, W. C. and Pless, V., Fundamentals of Error Correcting Codes (Cambridge University Press, Cambridge, 2003).CrossRefGoogle Scholar
Vinck, A. J. H., ‘Coded modulation for power line communications’, AEÜ J., 45–49, January 2000.Google Scholar