6 research outputs found
Permutation codes with specified packing radius
Most papers on permutation codes have concentrated on the minimum Hamming distance of the code. An (n, d) permutation code (or permutation array) is simply a set of permutations on n elements in which the Hamming distance between any pair of distinct permutations (or codewords) is at least d. An (n, 2e + 1) or (n, 2e +2) permutation code is able to correct up to e errors. These codes have a potential application to powerline communications. It is known that in an (n, 2e) permutation code the balls of radius e surrounding the codewords may all be pairwise disjoint, but usually some overlap. Thus an (n, 2e) permutation code is generally unable to correct e errors using nearest neighbour decoding. On the other hand, if the packing radius of the code is defined as the largest value of e for which the balls of radius e are all pairwise disjoint, a permutation code with packing radius e can be denoted by [n, e]. Such a permutation code can always correct e errors. In this paper it is shown that, in almost all cases considered, the number of codewords in the best [n, e] code found is substantially greater than the largest number of codewords in the best known (n, 2e + 1) code. Thus the packing radius more accurately specifies the requirement for an e-error-correcting permutation code than does the minimum Hamming distance. The techniques used include construction by automorphism group and several variations of clique search They are enhanced by two theoretical results which make the computations feasibl
New Bounds for Permutation Codes in Ulam Metric
New bounds on the cardinality of permutation codes equipped with the Ulam
distance are presented. First, an integer-programming upper bound is derived,
which improves on the Singleton-type upper bound in the literature for some
lengths. Second, several probabilistic lower bounds are developed, which
improve on the known lower bounds for large minimum distances. The results of a
computer search for permutation codes are also presented.Comment: To be presented at ISIT 2015, 5 page
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
Permutation codes with specified packing radius
Most papers on permutation codes have concentrated on the minimum Hamming distance of the code. An (n, d) permutation code (or permutation array) is simply a set of permutations on n elements in which the Hamming distance between any pair of distinct permutations (or codewords) is at least d. An (n, 2e + 1) or (n, 2e + 2) permutation code is able to correct up to e errors. These codes have a potential application to powerline communications. It is known that in an (n, 2e) permutation code the balls of radius e surrounding the codewords may all be pairwise disjoint, but usually some overlap. Thus an (n, 2e) permutation code is generally unable to correct e errors using nearest neighbour decoding. On the other hand, if the packing radius of the code is defined as the largest value of e for which the balls of radius e are all pairwise disjoint, a permutation code with packing radius e can be denoted by [n, e]. Such a permutation code can always correct e errors. In this paper it is shown that, in almost all cases considered, the number of codewords in the best [n, e] code found is substantially greater than the largest number of codewords in the best known (n, 2e + 1) code. Thus the packing radius more accurately specifies the requirement for an e-error-correcting permutation code than does the minimum Hamming distance. The techniques used include construction by automorphism group and several variations of clique search They are enhanced by two theoretical results which make the computations feasible