Neil I. Gillespie

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, while at the same time maintaining a constant power output. 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.

Uniqueness of certain completely regular Hadamard codes

Abstract 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. Moreover, we prove that these codes are completely transitive.

A note on binary completely regular codes with large minimum distance

Abstract We classify all binary error correcting completely regular codes of length n with minimum distance δ > n / 2 .

Conway groupoids and completely transitive codes

To each supersimple 2-(n,4,λ) design D one associates a ‘Conway groupoid’, which may be thought of as a natural generalisation of Conway’s Mathieu groupoid M13 which is constructed from P3.We show that Sp2m(2) and 22m. Sp2m(2) naturally occur as Conway groupoids associated to certain designs. It is shown that the incidence matrix associated to one of these designs generates a new family of completely transitive F2-linear codes with minimum distance 4 and covering radius 3, whereas the incidence matrix of the other design gives an alternative construction of a previously known family of completely transitive codes.We also give a new characterization of M13 and prove that, for a fixed λ > 0; there are finitely many Conway groupoids for which the set of morphisms does not contain all elements of the full alternating group.

Entry-faithful 2-neighbour transitive codes

We consider a code to be a subset of the vertex set of a Hamming graph. The set of

ELUSIVE CODES IN HAMMING GRAPHS

s

Alphabet-almost-simple 2-neighbour-transitive codes

s-neighbours of a code is the set of vertices, not in the code, at distance