5,812 research outputs found
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
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
Miscorrection probability beyond the minimum distance
The miscorrection probability of a list decoder is the probability that the decoder will have at least one non-causal codeword in its decoding sphere. Evaluating this probability is important when using a list-decoder as a conventional decoder since in that case we require the list to contain at most one codeword for most of the errors. A lower bound on the miscorrection is the main result. The key ingredient in the proof is a new combinatorial upper bound on the list-size for a general qāary block code. This bound is tighter than the best known on large alphabets, and it is shown to be very close to the algebraic bound for Reed-Solomon codes. Finally we discuss two known upper bounds on the miscorrection probability and unify them for linear MDS codes
Temporal Trends of Emergency Department Visits of Patients with Atrial Fibrillation:A Nationwide Population-Based Study
The question of list decoding error-correcting codes over finite fields (under the Hamming metric) has been widely studied in recent years. Motivated by the similar discrete linear structure of linear codes and point lattices in R N, and their many shared applications across complexity theory, cryptography, and coding theory, we initiate the study of list decoding for lattices. Namely: for a lattice L ā R N, given a target vector r ā R N and a distance parameter d, output the set of all lattice points w ā L that are within distance d of r. In this work we focus on combinatorial and algorithmic questions related to list decoding for the well-studied family of Barnes-Wall lattices. Our main contributions are twofold: 1. We give tight (up to polynomials) combinatorial bounds on the worst-case list size, showing it to be polynomial in the lattice dimension for any error radius bounded away from the latticeās minimum distance (in the Euclidean norm). 2. Building on the unique decoding algorithm of Micciancio and Nicolosi (ISIT ā08), we give a list-decoding algorithm that runs in time polynomial in the lattice dimension and worst-case list size, for any error radius. Moreover, our algorithm is highly parallelizable, and with sufficiently many processors can run in parallel time only poly-logarithmic in the lattice dimension. In particular, our results imply a polynomial-time list-decoding algorithm for any error radius bounded away from the minimum distance, thus beating a typical barrier for natural error-correcting codes posed by the Johnson radius
Local list decoding of homomorphisms
Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2006.Includes bibliographical references (leaves 47-49).We investigate the local-list decodability of codes whose codewords are group homomorphisms. The study of such codes was intiated by Goldreich and Levin with the seminal work on decoding the Hadamard code. Many of the recent abstractions of their initial algorithm focus on Locally Decodable Codes (LDC's) over finite fields. We derive our algorithmic approach from the list decoding of the Reed-Muller code over finite fields proposed by Sudan, Trevisan and Vadhan. Given an abelian group G and a fixed abelian group H, we give combinatorial bounds on the number of homomorphisms that have agreement 6 with an oracle-access function f : G --> H. Our bounds are polynomial in , where the degree of the polynomial depends on H. Also, depends on the distance parameter of the code, namely we consider to be slightly greater than 1-minimum distance. Furthermore, we give a local-list decoding algorithm for the homomorphisms that agree on a 3 fraction of the domain with a function f, the running time of which is poly(1/e, log G).by Elena Grigorescu.S.M
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
Problems on q-Analogs in Coding Theory
The interest in -analogs of codes and designs has been increased in the
last few years as a consequence of their new application in error-correction
for random network coding. There are many interesting theoretical, algebraic,
and combinatorial coding problems concerning these q-analogs which remained
unsolved. The first goal of this paper is to make a short summary of the large
amount of research which was done in the area mainly in the last few years and
to provide most of the relevant references. The second goal of this paper is to
present one hundred open questions and problems for future research, whose
solution will advance the knowledge in this area. The third goal of this paper
is to present and start some directions in solving some of these problems.Comment: arXiv admin note: text overlap with arXiv:0805.3528 by other author
On the Combinatorial Version of the Slepian-Wolf Problem
We study the following combinatorial version of the Slepian-Wolf coding
scheme. Two isolated Senders are given binary strings and respectively;
the length of each string is equal to , and the Hamming distance between the
strings is at most . The Senders compress their strings and
communicate the results to the Receiver. Then the Receiver must reconstruct
both strings and . The aim is to minimise the lengths of the transmitted
messages.
For an asymmetric variant of this problem (where one of the Senders transmits
the input string to the Receiver without compression) with deterministic
encoding a nontrivial lower bound was found by A.Orlitsky and K.Viswanathany.
In our paper we prove a new lower bound for the schemes with syndrome coding,
where at least one of the Senders uses linear encoding of the input string.
For the combinatorial Slepian-Wolf problem with randomized encoding the
theoretical optimum of communication complexity was recently found by the first
author, though effective protocols with optimal lengths of messages remained
unknown. We close this gap and present a polynomial time randomized protocol
that achieves the optimal communication complexity.Comment: 20 pages, 14 figures. Accepted to IEEE Transactions on Information
Theory (June 2018
On the List-Decodability of Random Linear Codes
For every fixed finite field \F_q, and , we
prove that with high probability a random subspace of \F_q^n of dimension
has the property that every Hamming ball of radius
has at most codewords.
This answers a basic open question concerning the list-decodability of linear
codes, showing that a list size of suffices to have rate within
of the "capacity" . Our result matches up to constant
factors the list-size achieved by general random codes, and gives an
exponential improvement over the best previously known list-size bound of
.
The main technical ingredient in our proof is a strong upper bound on the
probability that random vectors chosen from a Hamming ball centered at
the origin have too many (more than ) vectors from their linear
span also belong to the ball.Comment: 15 page
List decoding group homomorphisms between supersolvable groups
We show that the set of homomorphisms between two supersolvable groups can be
locally list decoded up to the minimum distance of the code, extending the
results of Dinur et al who studied the case where the groups are abelian.
Moreover, when specialized to the abelian case, our proof is more streamlined
and gives a better constant in the exponent of the list size. The constant is
improved from about 3.5 million to 105.Comment: 11 page
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