454 research outputs found
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
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
Characterisation of a family of neighbour transitive codes
We consider codes of length over an alphabet of size as subsets of
the vertex set of the Hamming graph . A code for which there
exists an automorphism group 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 that has a -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 , and for which the group has a non-trivial intersection with the
base group of . If is such a code, we show that, up to
equivalence, there exists a subcode that can be completely described,
and that either , or is a neighbour transitive frequency
permutation array and is the disjoint union of -translates of .
We also prove that any finite group can be identified in a natural way with a
neighbour transitive code.Comment: 30 Page
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