6 research outputs found
A characterization of the capacity of online (causal) binary channels
In the binary online (or "causal") channel coding model, a sender wishes to
communicate a message to a receiver by transmitting a codeword bit by bit via a channel limited to at most
corruptions. The channel is "online" in the sense that at the th step
of communication the channel decides whether to corrupt the th bit or not
based on its view so far, i.e., its decision depends only on the transmitted
bits . This is in contrast to the classical adversarial
channel in which the error is chosen by a channel that has a full knowledge on
the sent codeword .
In this work we study the capacity of binary online channels for two
corruption models: the {\em bit-flip} model in which the channel may flip at
most of the bits of the transmitted codeword, and the {\em erasure} model
in which the channel may erase at most bits of the transmitted codeword.
Specifically, for both error models we give a full characterization of the
capacity as a function of .
The online channel (in both the bit-flip and erasure case) has seen a number
of recent studies which present both upper and lower bounds on its capacity. In
this work, we present and analyze a coding scheme that improves on the
previously suggested lower bounds and matches the previously suggested upper
bounds thus implying a tight characterization
The benefit of a 1-bit jump-start, and the necessity of stochastic encoding, in jamming channels
We consider the problem of communicating a message in the presence of a
malicious jamming adversary (Calvin), who can erase an arbitrary set of up to
bits, out of transmitted bits . The capacity of such
a channel when Calvin is exactly causal, i.e. Calvin's decision of whether or
not to erase bit depends on his observations was
recently characterized to be . In this work we show two (perhaps)
surprising phenomena. Firstly, we demonstrate via a novel code construction
that if Calvin is delayed by even a single bit, i.e. Calvin's decision of
whether or not to erase bit depends only on (and
is independent of the "current bit" ) then the capacity increases to
when the encoder is allowed to be stochastic. Secondly, we show via a novel
jamming strategy for Calvin that, in the single-bit-delay setting, if the
encoding is deterministic (i.e. the transmitted codeword is a deterministic
function of the message ) then no rate asymptotically larger than is
possible with vanishing probability of error, hence stochastic encoding (using
private randomness at the encoder) is essential to achieve the capacity of
against a one-bit-delayed Calvin.Comment: 21 pages, 4 figures, extended draft of submission to ISIT 201
The Capacity of Online (Causal) -ary Error-Erasure Channels
In the -ary online (or "causal") channel coding model, a sender wishes to
communicate a message to a receiver by transmitting a codeword symbol by symbol via a channel
limited to at most errors and/or erasures. The channel is
"online" in the sense that at the th step of communication the channel
decides whether to corrupt the th symbol or not based on its view so far,
i.e., its decision depends only on the transmitted symbols .
This is in contrast to the classical adversarial channel in which the
corruption is chosen by a channel that has a full knowledge on the sent
codeword .
In this work we study the capacity of -ary online channels for a combined
corruption model, in which the channel may impose at most {\em errors} and
at most {\em erasures} on the transmitted codeword. The online
channel (in both the error and erasure case) has seen a number of recent
studies which present both upper and lower bounds on its capacity. In this
work, we give a full characterization of the capacity as a function of ,
and .Comment: This is a new version of the binary case, which can be found at
arXiv:1412.637
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