$2$-Neighbour-Transitive Codes with Small Blocks of Imprimitivity

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

doi:10.37236/8040

Cited by 5 publications

(7 citation statements)

References 23 publications

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“…The main result of [18] is stated below. The importance of this result, in the present context, is that it classifies all binary 2-neighbour-transitive codes with minimum distance at least 5 that have a repetition subcode.…”

Section: Proposition 22 [16 Proposition 25] Letmentioning

confidence: 99%

“…Neighbour-transitive codes are investigated in [19,20,21]. The class of 2-neighbour-transitive codes is the subject of [16,17,18], and the present work comprises part of the first author's PhD thesis [25]. Codes with 2-transitive actions on the entries of the Hamming graph have been used to find certain families of codes that achieve capacity on erasure channels [30], and many 2-neighbourtransitive codes indeed admit such actions (see Proposition 2.2).…”

Section: Introductionmentioning

confidence: 99%

“…As a consequence of these results, the proof of Theorem 1.2 amounts to finding all minimal subcodes of binary 2-neighbour-transitive and X-alphabetaffine codes, for some automorphism group X, having minimum distance at least 5. Note that such codes having a binary repetition subcode are classified in [18]. Table 1 gives certain parameters of the minimal binary 2-neighbour-transitive and Xalphabet-affine codes with minimum distance δ satisfying 5 δ m. While δ is not explicitly found in every case, bounds are given for those cases where δ is not known.…”

Section: Introductionmentioning

confidence: 99%

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Minimal binary 2-neighbour-transitive codes

Hawtin

Praeger

2020

Journal of Combinatorial Theory, Series A

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The main result here is a characterisation of binary 2-neighbour-transitive codes with minimum distance at least 5 via their minimal subcodes, which are found to be generated by certain designs. The motivation for studying this class of codes comes primarily from their relationship to the class of completely regular codes. The results contained here yield many more examples of 2-neighbour-transitive codes than previous classification results of families of 2-neighbour-transitive codes. In the process, new lower bounds on the minimum distance of particular sub-families are produced. Several results on the structure of 2-neighbour-transitive codes with arbitrary alphabet size are also proved. The proofs of the main results apply the classification of minimal and pre-minimal submodules of the permutation modules over F 2 for finite 2-transitive permutation groups. * The first author sincerely thanks Michael Giudici for reading first drafts of this work and is grateful for the support of an Australian Research Training Program Scholarship and a University of Western Australia Safety-Net Top-Up Scholarship. The research forms part of Australian Research Council Project FF0776186.1. (X, s)-neighbour-transitive if X acts transitively on each of the sets C, C 1 , . . . , C s , 2. X-neighbour-transitive if C is (X, 1)-neighbour-transitive, 3. X-completely transitive if C is (X, ρ)-neighbour-transitive, and, 4. s-neighbour-transitive, neighbour-transitive, or completely transitive, respectively, if C is (Aut(C), s)-neighbour-transitive, Aut(C)-neighbour-transitive, or Aut(C)-completely transitive, respectively.A variant of the above concept of complete transitivity was introduced for linear codes by Solé [33], with the above definition first appearing in [23]. Note that non-linear completely transitive codes do indeed exist; see [22]. Completely transitive codes form a subfamily of completely regular codes, and s-neighbour transitive codes are a sub-family of s-regular codes, for each s. It is hoped that studying 2-neighbour-transitive codes will lead to a better understanding of completely transitive and completely regular codes. Indeed a classification of 2-neighbour-transitive codes would have as a corollary a classification of completely transitive codes.The main result of the present paper, stated below, provides a characterisation of binary 2-neighbour-transitive codes with minimum distance at least 5. Theorem 1.2. Let C be a binary code in H(m, 2) with minimum distance at least 5. Then C is 2-neighbour-transitive if and only if one of the following holds:

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Section: Proposition 22 [16 Proposition 25] Letmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Minimal binary 2-neighbour-transitive codes

Hawtin

Praeger

2020

Journal of Combinatorial Theory, Series A

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“…In [16] they give an affirmative answer via the construction of an infinite family of examples. Similarly, the significance of the existence of s-elusive codes relates to the precise definition of s-neighbour-transitive codes (see [12,13,15,19]).…”

Section: Introductionmentioning

confidence: 99%

s-Elusive codes in Hamming graphs

Hawtin

2021

Des. Codes Cryptogr.

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A code is a subset of the vertex set of a Hamming graph. The set of s-neighbours of a code is the set of all vertices at Hamming distance s from their nearest codeword. A code C is s-elusive if there exists a distinct code C that is equivalent to C under the full automorphism group of the Hamming graph such that C and C have the same set of s-neighbours. We show that the minimum distance of an s-elusive code is at most 2s + 2, and that an s-elusive code with minimum distance at least 2s + 1 gives rise to a q-ary t-design with certain parameters. This leads to the construction of: an infinite family of 1-elusive and completely transitive codes, an infinite family of 2-elusive codes, and a single example of a 3-elusive code. Answers to several open questions on elusive codes are also provided.

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“…Neighbour-transitive codes in Hamming graphs were first studied in Gillespie's PhD thesis [17], as well as [21,22,23]. Investigating 2-neighbour-transitive codes in Hamming graphs constitutes a major part of the second author's PhD thesis [25]; see also [18,19,20,26]. Completely transitive codes are a subclass of completely regular codes (see [8]) and completely transitive codes in Hamming graphs have been the subject of [1,5,6,7,8,24].…”

Section: Introductionmentioning

confidence: 99%

Neighbour-Transitive Codes and Partial Spreads in Generalised Quadrangles

Crnković¹,

Hawtin²,

Švob³

2021

Preprint

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A code C in a generalised quadrangle Q is defined to be a subset of the vertex set of the point-line incidence graph Γ of Q. The minimum distance δ of C is the smallest distance between a pair of distinct elements of C. The graph metric gives rise to the distance partition {C, C 1 , . . . , C ρ }, where ρ is the maximum distance between any vertex of Γ and its nearest element of C. Since the diameter of Γ is 4, both ρ and δ are at most 4. If δ = 4 then C is a partial ovoid or partial spread of Q, and if, additionally, ρ = 2 then C is an ovoid or a spread. A code C in Q is neighbour-transitive if its automorphism group acts transitively on each of the sets C and C 1 . Our main results i) classify all neighbour-transitive codes admitting an insoluble group of automorphisms in thick classical generalised quadrangles that correspond to ovoids or spreads, and ii) give two infinite families and six sporadic examples of neighbourtransitive codes with minimum distance δ = 4 in the classical generalised quadrangle W 3 (q) that are not ovoids or spreads. * This work has been supported by the Croatian Science Foundation under the projects 6732 and 5713.1. C is equivalent to a regular spread if and only if C has covering radius 2.2. If |C| = q 2 then C has covering radius ρ = 4 and is equivalent to a spread minus a line.3. If |C| = q 2 − 1 then q = 2, 3, 5, 7 or 11 and C is equivalent to one of the codes in Example 5.4, each of which has covering radius ρ = 3.4. If |C| = q + 1 and C has covering radius ρ = 3 then C is equivalent to the set of points on a hyperbolic line.5. If q = 3 and |C| = 5 then C has covering radius ρ = 3 and is equivalent to the code given in Example 5.3.Note that an example of a code as in part 2 of Theorem 1.2 is given, for each q, by a regular spread of W 3 (q) with one line removed; see Lemma 4.3. We do not prove here that these are all such examples. However, since the automorphism group of such a code must have order divisible by q 2 (q + 1), [32, Table 1] suggests that perhaps this is the case. Hence, we pose the following question.

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$2$-Neighbour-Transitive Codes with Small Blocks of Imprimitivity

Cited by 5 publications

References 23 publications

Minimal binary 2-neighbour-transitive codes

Minimal binary 2-neighbour-transitive codes

s-Elusive codes in Hamming graphs

Neighbour-Transitive Codes and Partial Spreads in Generalised Quadrangles

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