15,913 research outputs found
Estimates on the Size of Symbol Weight Codes
The study of codes for powerlines communication has garnered much interest
over the past decade. Various types of codes such as permutation codes,
frequency permutation arrays, and constant composition codes have been proposed
over the years. In this work we study a type of code called the bounded symbol
weight codes which was first introduced by Versfeld et al. in 2005, and a
related family of codes that we term constant symbol weight codes. We provide
new upper and lower bounds on the size of bounded symbol weight and constant
symbol weight codes. We also give direct and recursive constructions of codes
for certain parameters.Comment: 14 pages, 4 figure
Coding for Errors and Erasures in Random Network Coding
The problem of error-control in random linear network coding is considered. A
``noncoherent'' or ``channel oblivious'' model is assumed where neither
transmitter nor receiver is assumed to have knowledge of the channel transfer
characteristic. Motivated by the property that linear network coding is
vector-space preserving, information transmission is modelled as the injection
into the network of a basis for a vector space and the collection by the
receiver of a basis for a vector space . A metric on the projective geometry
associated with the packet space is introduced, and it is shown that a minimum
distance decoder for this metric achieves correct decoding if the dimension of
the space is sufficiently large. If the dimension of each codeword
is restricted to a fixed integer, the code forms a subset of a finite-field
Grassmannian, or, equivalently, a subset of the vertices of the corresponding
Grassmann graph. Sphere-packing and sphere-covering bounds as well as a
generalization of the Singleton bound are provided for such codes. Finally, a
Reed-Solomon-like code construction, related to Gabidulin's construction of
maximum rank-distance codes, is described and a Sudan-style ``list-1'' minimum
distance decoding algorithm is provided.Comment: This revised paper contains some minor changes and clarification
A Rank-Metric Approach to Error Control in Random Network Coding
The problem of error control in random linear network coding is addressed
from a matrix perspective that is closely related to the subspace perspective
of K\"otter and Kschischang. A large class of constant-dimension subspace codes
is investigated. It is shown that codes in this class can be easily constructed
from rank-metric codes, while preserving their distance properties. Moreover,
it is shown that minimum distance decoding of such subspace codes can be
reformulated as a generalized decoding problem for rank-metric codes where
partial information about the error is available. This partial information may
be in the form of erasures (knowledge of an error location but not its value)
and deviations (knowledge of an error value but not its location). Taking
erasures and deviations into account (when they occur) strictly increases the
error correction capability of a code: if erasures and
deviations occur, then errors of rank can always be corrected provided that
, where is the minimum rank distance of the
code. For Gabidulin codes, an important family of maximum rank distance codes,
an efficient decoding algorithm is proposed that can properly exploit erasures
and deviations. In a network coding application where packets of length
over are transmitted, the complexity of the decoding algorithm is given
by operations in an extension field .Comment: Minor corrections; 42 pages, to be published at the IEEE Transactions
on Information Theor
Product Construction of Affine Codes
Binary matrix codes with restricted row and column weights are a desirable
method of coded modulation for power line communication. In this work, we
construct such matrix codes that are obtained as products of affine codes -
cosets of binary linear codes. Additionally, the constructions have the
property that they are systematic. Subsequently, we generalize our construction
to irregular product of affine codes, where the component codes are affine
codes of different rates.Comment: 13 pages, to appear in SIAM Journal on Discrete Mathematic
Generalized List Decoding
This paper concerns itself with the question of list decoding for general
adversarial channels, e.g., bit-flip () channels, erasure
channels, (-) channels, channels, real adder
channels, noisy typewriter channels, etc. We precisely characterize when
exponential-sized (or positive rate) -list decodable codes (where the
list size is a universal constant) exist for such channels. Our criterion
asserts that:
"For any given general adversarial channel, it is possible to construct
positive rate -list decodable codes if and only if the set of completely
positive tensors of order- with admissible marginals is not entirely
contained in the order- confusability set associated to the channel."
The sufficiency is shown via random code construction (combined with
expurgation or time-sharing). The necessity is shown by
1. extracting equicoupled subcodes (generalization of equidistant code) from
any large code sequence using hypergraph Ramsey's theorem, and
2. significantly extending the classic Plotkin bound in coding theory to list
decoding for general channels using duality between the completely positive
tensor cone and the copositive tensor cone. In the proof, we also obtain a new
fact regarding asymmetry of joint distributions, which be may of independent
interest.
Other results include
1. List decoding capacity with asymptotically large for general
adversarial channels;
2. A tight list size bound for most constant composition codes
(generalization of constant weight codes);
3. Rederivation and demystification of Blinovsky's [Bli86] characterization
of the list decoding Plotkin points (threshold at which large codes are
impossible);
4. Evaluation of general bounds ([WBBJ]) for unique decoding in the error
correction code setting
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