### Abstract

A code C in the Hamming graph Γ = H(m, q) is a subset of the vertex set V Γ of the Hamming graph; the elements of C are called codewords. Any such code C induces a partition {C, C_{1}, …, C_{ρ}} of V Γ, where ρ is the covering radius of the code, based on the distance each vertex is to its nearest codeword. For s ∈ {1, …, ρ} and X ≤ Aut(C), if X is transitive on each of C, C_{1}, …, C_{s}, then C is said to be (X, s)-neighbour-transitive. In particular, C is said to be X-completely transitive if C is (X, ρ)-neighbour-transitive. It is known that for any (X, 2)-neighbour-transitive code with minimum distance at least 5, either i) X is faithful on the set of coordinate entries, ii) C is X-alphabet-almost-simple or iii) C is X-alphabet-affine. Classifications of (X, 2)-neighbour-transitive codes in the first two categories having minimum distance at least 5 and 3, respectively, have been achieved in previous papers. Hence this paper considers case iii). Let q = p^{dm} and identify the vertex set of H(m, q) with F^{dm} p. The main result of this paper classifies (X, 2)-neighbour-transitive codes with minimum distance at least 5 that contain, as a block of imrimitivity for the action of X on C, an F_{p}-subspace of F^{dm} p of dimension at most d. When considering codes with minimum distance at least 5, X-completely transitive codes are a proper subclass of (X, 2)-neighbour-transitive codes. This leads, as a corollary of the main result, to a solution of a problem posed by Giudici in 1998 on completely transitive codes.

Original language | English |
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Article number | P1.42 |

Number of pages | 16 |

Journal | Electronic Journal of Combinatorics |

Volume | 27 |

Issue number | 1 |

Early online date | 29 Jan 2020 |

DOIs | |

Publication status | Published - 7 Feb 2020 |

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## Cite this

*Electronic Journal of Combinatorics*,

*27*(1), [P1.42]. https://doi.org/10.37236/8040