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ELUSIVE CODES IN HAMMING GRAPHS

  • DANIEL R. HAWTIN (a1), NEIL I. GILLESPIE (a1) and CHERYL E. PRAEGER (a1) (a2)
Abstract

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.

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References
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[1]Bailey, R. F., ‘Error-correcting codes from permutation groups’, Discrete Math. 309 (2009), 42534265.
[2]Blake, I. F., ‘Permutation codes for discrete channels’, IEEE Trans. Inform. Theory 20 (1974), 138140.
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[5]Chu, W., Colbourn, C. J. and Dukes, P., ‘Constructions for permutation codes in powerline communications’, Des. Codes Cryptogr. 32 (1–3) (May 2004), 5164.
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[7]Gillespie, N. I. and Praeger, C. E., ‘From neighbour transitive codes to frequency permutation arrays’. 2012. arXiv:1204.2900v1.
[8]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.
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[11]Vinck, A. J. H., ‘Coded modulation for power line communications’, AEÜ J., 45–49, January 2000.
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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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