4,771 research outputs found
Upper bound on list-decoding radius of binary codes
Consider the problem of packing Hamming balls of a given relative radius
subject to the constraint that they cover any point of the ambient Hamming
space with multiplicity at most . For odd an asymptotic upper bound
on the rate of any such packing is proven. Resulting bound improves the best
known bound (due to Blinovsky'1986) for rates below a certain threshold. Method
is a superposition of the linear-programming idea of Ashikhmin, Barg and Litsyn
(that was used previously to improve the estimates of Blinovsky for ) and
a Ramsey-theoretic technique of Blinovsky. As an application it is shown that
for all odd the slope of the rate-radius tradeoff is zero at zero rate.Comment: IEEE Trans. Inform. Theory, accepte
Combinatorial limitations of average-radius list-decoding
We study certain combinatorial aspects of list-decoding, motivated by the
exponential gap between the known upper bound (of ) and lower
bound (of ) for the list-size needed to decode up to
radius with rate away from capacity, i.e., 1-\h(p)-\gamma (here
and ). Our main result is the following:
We prove that in any binary code of rate
1-\h(p)-\gamma, there must exist a set of
codewords such that the average distance of the
points in from their centroid is at most . In other words,
there must exist codewords with low "average
radius." The standard notion of list-decoding corresponds to working with the
maximum distance of a collection of codewords from a center instead of average
distance. The average-radius form is in itself quite natural and is implied by
the classical Johnson bound.
The remaining results concern the standard notion of list-decoding, and help
clarify the combinatorial landscape of list-decoding:
1. We give a short simple proof, over all fixed alphabets, of the
above-mentioned lower bound. Earlier, this bound
followed from a complicated, more general result of Blinovsky.
2. We show that one {\em cannot} improve the
lower bound via techniques based on identifying the zero-rate regime for list
decoding of constant-weight codes.
3. We show a "reverse connection" showing that constant-weight codes for list
decoding imply general codes for list decoding with higher rate.
4. We give simple second moment based proofs of tight (up to constant
factors) lower bounds on the list-size needed for list decoding random codes
and random linear codes from errors as well as erasures.Comment: 28 pages. Extended abstract in RANDOM 201
List Decoding Tensor Products and Interleaved Codes
We design the first efficient algorithms and prove new combinatorial bounds
for list decoding tensor products of codes and interleaved codes. We show that
for {\em every} code, the ratio of its list decoding radius to its minimum
distance stays unchanged under the tensor product operation (rather than
squaring, as one might expect). This gives the first efficient list decoders
and new combinatorial bounds for some natural codes including multivariate
polynomials where the degree in each variable is bounded. We show that for {\em
every} code, its list decoding radius remains unchanged under -wise
interleaving for an integer . This generalizes a recent result of Dinur et
al \cite{DGKS}, who proved such a result for interleaved Hadamard codes
(equivalently, linear transformations). Using the notion of generalized Hamming
weights, we give better list size bounds for {\em both} tensoring and
interleaving of binary linear codes. By analyzing the weight distribution of
these codes, we reduce the task of bounding the list size to bounding the
number of close-by low-rank codewords. For decoding linear transformations,
using rank-reduction together with other ideas, we obtain list size bounds that
are tight over small fields.Comment: 32 page
A Combinatorial Bound on the List Size
In this paper we study the scenario in which a server sends dynamic data over a single broadcast channel to
a number of passive clients. We consider the data to consist of discrete packets, where each update is sent in a
separate packet. On demand, each client listens to the channel in order to obtain the most recent data packet. Such
scenarios arise in many practical applications such as the distribution of weather and traffic updates to wireless
mobile devices and broadcasting stock price information over the Internet.
To satisfy a request, a client must listen to at least one packet from beginning to end. We thus consider the design
of a broadcast schedule which minimizes the time that passes between a clients request and the time that it hears a new data packet, i.e., the waiting time of the client. Previous studies have addressed this objective, assuming that client requests are distributed uniformly over time. However, in the general setting, the clients behavior is difficult to predict and might not be known to the server. In this work we consider the design of universal schedules that guarantee a short waiting time for any possible client behavior. We define the model of dynamic broadcasting in
the universal setting, and prove various results regarding the waiting time achievable in this framework
Two Theorems in List Decoding
We prove the following results concerning the list decoding of
error-correcting codes:
(i) We show that for \textit{any} code with a relative distance of
(over a large enough alphabet), the following result holds for \textit{random
errors}: With high probability, for a \rho\le \delta -\eps fraction of random
errors (for any \eps>0), the received word will have only the transmitted
codeword in a Hamming ball of radius around it. Thus, for random errors,
one can correct twice the number of errors uniquely correctable from worst-case
errors for any code. A variant of our result also gives a simple algorithm to
decode Reed-Solomon codes from random errors that, to the best of our
knowledge, runs faster than known algorithms for certain ranges of parameters.
(ii) We show that concatenated codes can achieve the list decoding capacity
for erasures. A similar result for worst-case errors was proven by Guruswami
and Rudra (SODA 08), although their result does not directly imply our result.
Our results show that a subset of the random ensemble of codes considered by
Guruswami and Rudra also achieve the list decoding capacity for erasures.
Our proofs employ simple counting and probabilistic arguments.Comment: 19 pages, 0 figure
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
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