3,169 research outputs found
Achieving the Uniform Rate Region of General Multiple Access Channels by Polar Coding
We consider the problem of polar coding for transmission over -user
multiple access channels. In the proposed scheme, all users encode their
messages using a polar encoder, while a multi-user successive cancellation
decoder is deployed at the receiver. The encoding is done separately across the
users and is independent of the target achievable rate. For the code
construction, the positions of information bits and frozen bits for each of the
users are decided jointly. This is done by treating the polar transformations
across all the users as a single polar transformation with a certain
\emph{polarization base}. We characterize the resolution of achievable rates on
the dominant face of the uniform rate region in terms of the number of users
and the length of the polarization base . In particular, we prove that
for any target rate on the dominant face, there exists an achievable rate, also
on the dominant face, within the distance at most
from the target rate. We then prove that the proposed MAC polar coding scheme
achieves the whole uniform rate region with fine enough resolution by changing
the decoding order in the multi-user successive cancellation decoder, as
and the code block length grow large. The encoding and decoding
complexities are and the asymptotic block error probability of
is guaranteed. Examples of achievable rates for
the -user multiple access channel are provided
The covering radius problem for sets of perfect matchings
Consider the family of all perfect matchings of the complete graph
with vertices. Given any collection of perfect matchings of
size , there exists a maximum number such that if ,
then there exists a perfect matching that agrees with each perfect matching in
in at most edges. We use probabilistic arguments to give
several lower bounds for . We also apply the Lov\'asz local lemma to
find a function such that if each edge appears at most times
then there exists a perfect matching that agrees with each perfect matching in
in at most edges. This is an analogue of an extremal result
vis-\'a-vis the covering radius of sets of permutations, which was studied by
Cameron and Wanless (cf. \cite{cameron}), and Keevash and Ku (cf. \cite{ku}).
We also conclude with a conjecture of a more general problem in hypergraph
matchings.Comment: 10 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
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
- …