21 research outputs found
Elusive Codes in Hamming Graphs
We consider a code to be a subset of the vertex set of a Hamming graph. We
examine elusive pairs, code-group pairs where the code is not determined by
knowledge of its set of neighbours. We construct a new infinite family of
elusive pairs, where the group in question acts transitively on the set of
neighbours of the code. In our examples, we find that the alphabet size always
divides the length of the code, and prove that there is no elusive pair for the
smallest set of parameters for which this is not the case. We also pose several
questions regarding elusive pairs
On Optimal Anticodes over Permutations with the Infinity Norm
Motivated by the set-antiset method for codes over permutations under the
infinity norm, we study anticodes under this metric. For half of the parameter
range we classify all the optimal anticodes, which is equivalent to finding the
maximum permanent of certain -matrices. For the rest of the cases we
show constraints on the structure of optimal anticodes
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
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.Comment: 20 page
A new table of permutation codes
Permutation codes (or permutation arrays) have received considerable interest in recent years, partly motivated by a potential application to powerline communication. Powerline communication is the transmission of data over the electricity distribution system. This environment is rather hostile to communication and the requirements are such that permutation codes may be suitable. The problem addressed in this study is the construction of permutation codes with a specified length and minimum Hamming distance, and with as many codewords (permutations) as possible. A number of techniques are used including construction by automorphism group and several variations of clique search based on vertex degrees. Many significant improvements are obtained to the size of the best known code
Uncoverings on graphs and network reliability
We propose a network protocol similar to the -tree protocol of Itai and
Rodeh [{\em Inform.\ and Comput.}\ {\bf 79} (1988), 43--59]. To do this, we
define an {\em -uncovering-by-bases} for a connected graph to be a
collection of spanning trees for such that any -subset of
edges of is disjoint from at least one tree in , where is
some integer strictly less than the edge connectivity of . We construct
examples of these for some infinite families of graphs. Many of these infinite
families utilise factorisations or decompositions of graphs. In every case the
size of the uncovering-by-bases is no larger than the number of edges in the
graph and we conjecture that this may be true in general.Comment: 12 pages, 5 figure
Permutation codes
AbstractThere are many analogies between subsets and permutations of a set, and in particular between sets of subsets and sets of permutations. The theories share many features, but there are also big differences. This paper is a survey of old and new results about sets (and groups) of permutations, concentrating on the analogies and on the relations to coding theory. Several open problems are described
Decoding generalised hyperoctahedral groups and asymptotic analysis of correctible error patterns
We demonstrate a majority-logic decoding algorithm for decoding the
generalised hyperoctahedral group when thought of as
an error-correcting code. We also find the complexity of this
decoding algorithm and compare it with that of another, more general, algorithm.
Finally, we enumerate the number of error patterns exceeding the
correction capability that can be successfully decoded by this
algorithm, and analyse this asymptotically