2-neighbour-transitive codes with small blocks of imprimitivity

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

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)

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
JournalElectronic Journal of Combinatorics
Volume27
Issue number1
DOIs
Publication statusPublished - 2020

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