Classification of a family of completely transitive codes

Gillespie, Neil I.; Giudici, Michael; Praeger, Cheryl E.

doi:10.48550/arxiv.1208.0393

Cited by 4 publications

(8 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It is a consequence of the definitions that neighbour transitive codes are necessarily 1 -regular, and it is known that completely transitive codes are completely regular [25]. The next result follows directly from [24,Lemma 2.13].…”

Section: Definitions and Preliminariesmentioning

confidence: 83%

“…It has minimum distance δ = m and is one of the simplest neighbour transitive codes [20]. It is also true that Rep(m, 2) is completely transitive [24]. However, if m 4 and q 3 then Rep(m, q) is not completely transitive [24,Lemma 2.15].…”

Section: Definitions and Preliminariesmentioning

confidence: 99%

“…It is also true that Rep(m, 2) is completely transitive [24]. However, if m 4 and q 3 then Rep(m, q) is not completely transitive [24,Lemma 2.15].…”

Section: Definitions and Preliminariesmentioning

confidence: 99%

“…A result of Brouwer et al [8, p.353] shows that certain families of distance-regular graphs are coset graphs of additive completely regular codes, and so a classification of these codes may be of interest to graph theorists as well as coding theorists. With this in mind, a classification of completely transitive codes seems valuable, and as such the authors have classified all X -completely transitive codes with K := X ∩ B = 1 and minimum distance at least 5 [24]. The case K = 1 is more difficult, however.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Characterisation of a family of neighbour transitive codes

Gillespie,

Praeger

2014

Preprint

Self Cite

View full text Add to dashboard Cite

We consider codes of length m over an alphabet of size q as subsets of the vertex set of the Hamming graph Γ = H(m, q) . A code for which there exists an automorphism group X Aut(Γ) that acts transitively on the code and on its set of neighbours is said to be neighbour transitive, and were introduced by the authors as a group theoretic analogue to the assumption that single errors are equally likely over a noisy channel. Examples of neighbour transitive codes include the Hamming codes, various Golay codes, certain Hadamard codes, the Nordstrom Robinson codes, certain permutation codes and frequency permutation arrays, which have connections with powerline communication, and also completely transitive codes, a subfamily of completely regular codes, which themselves have attracted a lot of interest. It is known that for any neighbour transitive code with minimum distance at least 3 there exists a subgroup of X that has a 2 -transitive action on the alphabet over which the code is defined.Therefore, by Burnside's theorem, this action is of almost simple or affine type. If the action is of almost simple type, we say the code is alphabet almost simple neighbour transitive. In this paper we characterise a family of neighbour transitive codes, in particular, the alphabet almost simple neighbour transitive codes with minimum distance at least 3 , and for which the group X has a non-trivial intersection with the base group of Aut(Γ) . If C is such a code, we show that, up to equivalence, there exists a subcode ∆ that can be completely described, and that either C = ∆ , or ∆ is a neighbour transitive frequency permutation array and C is the disjoint union of X -translates of ∆ .We also prove that any finite group can be identified in a natural way with a neighbour transitive code.

show abstract

Section: Definitions and Preliminariesmentioning

confidence: 83%

Section: Definitions and Preliminariesmentioning

confidence: 99%

“…It is also true that Rep(m, 2) is completely transitive [24]. However, if m 4 and q 3 then Rep(m, q) is not completely transitive [24,Lemma 2.15].…”

Section: Definitions and Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Characterisation of a family of neighbour transitive codes

Gillespie,

Praeger

2014

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…This led the author to prove Theorem 1.1, and in particular, give a proof that is independent of [1]. Furthermore, this result plays an essential role in the classification of another family of completely regular codes [6].…”

Section: Introductionmentioning

confidence: 94%

A note on binary completely regular codes with large minimum distance

Gillespie

2013

Discrete Mathematics

Self Cite

View full text Add to dashboard Cite

show abstract

Entry-faithful 2-neighbour transitive codes

Gillespie

Giudici

Hawtin

et al. 2015

Des. Codes Cryptogr.

Self Cite

View full text Add to dashboard Cite

Abstract. We consider a code to be a subset of the vertex set of a Hamming graph. The set of s-neighbours of a code is the set of vertices, not in the code, at distance s from some codeword, but not distance less than s from any codeword. A 2-neighbour transitive code is a code which admits a group X of automorphisms which is transitive on the s-neighbours, for s = 1, 2, and transitive on the code itself. We give a classification of 2-neighbour transitive codes, with minimum distance δ 5, for which X acts faithfully on the set of entries of the Hamming graph.

show abstract

Classification of a family of completely transitive codes

Cited by 4 publications

References 18 publications

Characterisation of a family of neighbour transitive codes

Characterisation of a family of neighbour transitive codes

A note on binary completely regular codes with large minimum distance

Entry-faithful 2-neighbour transitive codes

Contact Info

Product

Resources

About