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Neil I. Gillespie

Publications and source records attributed to Neil I. Gillespie.

24 records · Page 2Linked to original sources
arXiv2014

Diagonally Neighbour Transitive Codes and Frequency Permutation Arrays

Constant composition codes have been proposed as suitable coding schemes to solve the narrow band and impulse noise problems associated with powerline communication. In particular, a certain class of constant composition codes called frequency permutation arrays have been suggested as ideal, in some sense, for these purposes. In this paper we characterise a family of neighbour transitive codes in Hamming graphs in which frequency permutation arrays play a central rode. We also classify all the permutation codes generated by groups in this family.

math.CO↗
arXiv2014

Twisted Permutation Codes

We introduce twisted permutation codes, which are frequency permutation arrays analogous to repetition permutation codes, namely, codes obtained from the repetition construction applied to a permutation code. In particular, we show that a lower bound for the minimum distance of a twisted permutation code is the minimum distance of a repetition permutation code. We give examples where this bound is tight, but more importantly, we give examples of twisted permutation codes with minimum distance strictly greater than this lower bound.

math.CO↗
arXiv2012

Uniqueness of certain completely regular Hadamard codes

We classify binary completely regular codes of length $m$ with minimum distance $δ$ for $(m,δ)=(12,6)$ and $(11,5)$. We prove that such codes are unique up to equivalence, and in particular, are equivalent to certain Hadamard codes. We prove that the automorphism groups of these Hadamard codes, modulo the kernel of a particular action, are isomorphic to certain Mathieu groups, from which we prove that completely regular codes with these parameters are necessarily completely transitive.

math.CO↗
arXiv2012

Classification of a family of completely transitive codes

The completely regular codes in Hamming graphs have a high degree of combinatorial symmetry and have attracted a lot of interest since their introduction in 1973 by Delsarte. This paper studies the subfamily of completely transitive codes, those in which an automorphism group is transitive on each part of the distance partition. This family is a natural generalisation of the binary completely transitive codes introduced by Sole in 1990. We take the first step towards a classification of these codes, determining those for which the automorphism group is faithful on entries.

math.CO↗
arXiv2011

Neighbour transitivity on codes in Hamming graphs

We consider a \emph{code} to be a subset of the vertex set of a \emph{Hamming graph}. In this setting a \emph{neighbour} of the code is a vertex which differs in exactly one entry from some codeword. This paper examines codes with the property that some group of automorphisms acts transitively on the \emph{set of neighbours} of the code. We call these codes \emph{neighbour transitive}. We obtain sufficient conditions for a neighbour transitive group to fix the code setwise. Moreover, we construct an infinite family of neighbour transitive codes, with \emph{minimum distance} $δ=4$, where this is not the case. That is to say, knowledge of even the complete set of code neighbours does not determine the code.

math.CO↗