2-neighbour-transitive codes with small blocks of imprimitivity

Neil I. Gillespie, Daniel R. Hawtin, Cheryl E. Praeger

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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, C1, …, Cρ} 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, C1, …, Cs, 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 = pdm and identify the vertex set of H(m, q) with Fdm 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 Fp-subspace of Fdm 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 languageEnglish
Article numberP1.42
Number of pages16
JournalElectronic Journal of Combinatorics
Volume27
Issue number1
Early online date29 Jan 2020
DOIs
Publication statusPublished - 7 Feb 2020

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