58 research outputs found
List Decodability at Small Radii
, the smallest for which every binary error-correcting code
of length and minimum distance is decodable with a list of size
up to radius , is determined for all . As a result,
is determined for all , except for 42 values of .Comment: to appear in Designs, Codes, and Cryptography (accepted October 2010
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
List decoding Reed-Muller codes over small fields
The list decoding problem for a code asks for the maximal radius up to which
any ball of that radius contains only a constant number of codewords. The list
decoding radius is not well understood even for well studied codes, like
Reed-Solomon or Reed-Muller codes.
Fix a finite field . The Reed-Muller code
is defined by -variate degree-
polynomials over . In this work, we study the list decoding radius
of Reed-Muller codes over a constant prime field ,
constant degree and large . We show that the list decoding radius is
equal to the minimal distance of the code.
That is, if we denote by the normalized minimal distance of
, then the number of codewords in any ball of
radius is bounded by independent
of . This resolves a conjecture of Gopalan-Klivans-Zuckerman [STOC 2008],
who among other results proved it in the special case of
; and extends the work of Gopalan [FOCS 2010] who
proved the conjecture in the case of .
We also analyse the number of codewords in balls of radius exceeding the
minimal distance of the code. For , we show that the number of
codewords of in a ball of radius is bounded by , where
is independent of . The dependence on is tight.
This extends the work of Kaufman-Lovett-Porat [IEEE Inf. Theory 2012] who
proved similar bounds over .
The proof relies on several new ingredients: an extension of the
Frieze-Kannan weak regularity to general function spaces, higher-order Fourier
analysis, and an extension of the Schwartz-Zippel lemma to compositions of
polynomials.Comment: fixed a bug in the proof of claim 5.6 (now lemma 5.5
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
Multiple Packing: Lower and Upper Bounds
We study the problem of high-dimensional multiple packing in Euclidean space.
Multiple packing is a natural generalization of sphere packing and is defined
as follows. Let and . A multiple packing is a
set of points in such that any point in lies in the intersection of at most balls of radius around points in . We study the multiple packing
problem for both bounded point sets whose points have norm at most
for some constant and unbounded point sets whose points are allowed to be
anywhere in . Given a well-known connection with coding theory,
multiple packings can be viewed as the Euclidean analog of list-decodable
codes, which are well-studied for finite fields. In this paper, we derive
various bounds on the largest possible density of a multiple packing in both
bounded and unbounded settings. A related notion called average-radius multiple
packing is also studied. Some of our lower bounds exactly pin down the
asymptotics of certain ensembles of average-radius list-decodable codes, e.g.,
(expurgated) Gaussian codes and (expurgated) spherical codes. In particular,
our lower bound obtained from spherical codes is the best known lower bound on
the optimal multiple packing density and is the first lower bound that
approaches the known large limit under the average-radius notion of
multiple packing. To derive these results, we apply tools from high-dimensional
geometry and large deviation theory.Comment: The paper arXiv:2107.05161 has been split into three parts with new
results added and significant revision. This paper is one of the three parts.
The other two are arXiv:2211.04408 and arXiv:2211.0440
List Decoding Random Euclidean Codes and Infinite Constellations
We study the list decodability of different ensembles of codes over the real
alphabet under the assumption of an omniscient adversary. It is a well-known
result that when the source and the adversary have power constraints and
respectively, the list decoding capacity is equal to . Random spherical codes achieve constant list
sizes, and the goal of the present paper is to obtain a better understanding of
the smallest achievable list size as a function of the gap to capacity. We show
a reduction from arbitrary codes to spherical codes, and derive a lower bound
on the list size of typical random spherical codes. We also give an upper bound
on the list size achievable using nested Construction-A lattices and infinite
Construction-A lattices. We then define and study a class of infinite
constellations that generalize Construction-A lattices and prove upper and
lower bounds for the same. Other goodness properties such as packing goodness
and AWGN goodness of infinite constellations are proved along the way. Finally,
we consider random lattices sampled from the Haar distribution and show that if
a certain number-theoretic conjecture is true, then the list size grows as a
polynomial function of the gap-to-capacity
Smoothing of binary codes, uniform distributions, and applications
The action of a noise operator on a code transforms it into a distribution on
the respective space. Some common examples from information theory include
Bernoulli noise acting on a code in the Hamming space and Gaussian noise acting
on a lattice in the Euclidean space. We aim to characterize the cases when the
output distribution is close to the uniform distribution on the space, as
measured by R{\'e}nyi divergence of order . A version of
this question is known as the channel resolvability problem in information
theory, and it has implications for security guarantees in wiretap channels,
error correction, discrepancy, worst-to-average case complexity reductions, and
many other problems.
Our work quantifies the requirements for asymptotic uniformity (perfect
smoothing) and identifies explicit code families that achieve it under the
action of the Bernoulli and ball noise operators on the code. We derive
expressions for the minimum rate of codes required to attain asymptotically
perfect smoothing. In proving our results, we leverage recent results from
harmonic analysis of functions on the Hamming space. Another result pertains to
the use of code families in Wyner's transmission scheme on the binary wiretap
channel. We identify explicit families that guarantee strong secrecy when
applied in this scheme, showing that nested Reed-Muller codes can transmit
messages reliably and securely over a binary symmetric wiretap channel with a
positive rate. Finally, we establish a connection between smoothing and error
correction in the binary symmetric channel
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