605 research outputs found
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
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
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
New characterisations of the Nordstrom–Robinson codes
In his doctoral thesis, Snover proved that any binary code
is equivalent to the Nordstrom-Robinson code or the punctured
Nordstrom-Robinson code for or respectively. We
prove that these codes are also characterised as \emph{completely regular}
binary codes with or , and moreover, that they are
\emph{completely transitive}. Also, it is known that completely transitive
codes are necessarily completely regular, but whether the converse holds has up
to now been an open question. We answer this by proving that certain completely
regular codes are not completely transitive, namely, the (Punctured) Preparata
codes other than the (Punctured) Nordstrom-Robinson code
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