700 research outputs found
Families of nested completely regular codes and distance-regular graphs
In this paper infinite families of linear binary nested completely regular
codes are constructed. They have covering radius equal to or ,
and are -th parts, for of binary (respectively,
extended binary) Hamming codes of length (respectively, ), where
. In the usual way, i.e., as coset graphs, infinite families of embedded
distance-regular coset graphs of diameter equal to or are
constructed. In some cases, the constructed codes are also completely
transitive codes and the corresponding coset graphs are distance-transitive
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.Comment: 16 page
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} , where this is not
the case. That is to say, knowledge of even the complete set of code neighbours
does not determine the code
Distance-regular graphs
This is a survey of distance-regular graphs. We present an introduction to
distance-regular graphs for the reader who is unfamiliar with the subject, and
then give an overview of some developments in the area of distance-regular
graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A.,
Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page
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