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Neighbour transitivity on codes in Hamming graphs

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Abstract

We consider a code to be a subset of the vertex set of a Hamming graph. In this setting a neighbour of the code is a vertex which differs in exactly one entry from some codeword. This paper examines codes with the property that some group of automorphisms acts transitively on the set of neighbours of the code. We call these codes neighbour transitive. We obtain sufficient conditions for a neighbour transitive group to fix the code setwise. Moreover, we construct an infinite family of neighbour transitive codes, with minimum distance δ = 4, where this is not the case. That is to say, knowledge of even the complete set of code neighbours does not determine the code.

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Authors and Affiliations

  1. Centre for the Mathematics of Symmetry and Computation, The University of Western Australia, 35 Stirling Highway, Crawley, Perth, WA, 6009, Australia

    Neil I. Gillespie & Cheryl E. Praeger

  2. King Abdulaziz University, Jeddah, Saudi Arabia

    Cheryl E. Praeger

Authors
  1. Neil I. Gillespie
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  2. Cheryl E. Praeger
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Correspondence to Neil I. Gillespie.

Additional information

Communicated by J. Bierbrauer.

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Gillespie, N.I., Praeger, C.E. Neighbour transitivity on codes in Hamming graphs. Des. Codes Cryptogr. 67, 385–393 (2013). https://doi.org/10.1007/s10623-012-9614-5

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  • DOI: https://doi.org/10.1007/s10623-012-9614-5

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