52 research outputs found
Channel Detection in Coded Communication
We consider the problem of block-coded communication, where in each block,
the channel law belongs to one of two disjoint sets. The decoder is aimed to
decode only messages that have undergone a channel from one of the sets, and
thus has to detect the set which contains the prevailing channel. We begin with
the simplified case where each of the sets is a singleton. For any given code,
we derive the optimum detection/decoding rule in the sense of the best
trade-off among the probabilities of decoding error, false alarm, and
misdetection, and also introduce sub-optimal detection/decoding rules which are
simpler to implement. Then, various achievable bounds on the error exponents
are derived, including the exact single-letter characterization of the random
coding exponents for the optimal detector/decoder. We then extend the random
coding analysis to general sets of channels, and show that there exists a
universal detector/decoder which performs asymptotically as well as the optimal
detector/decoder, when tuned to detect a channel from a specific pair of
channels. The case of a pair of binary symmetric channels is discussed in
detail.Comment: Submitted to IEEE Transactions on Information Theor
On the Reliability Function of Distributed Hypothesis Testing Under Optimal Detection
The distributed hypothesis testing problem with full side-information is
studied. The trade-off (reliability function) between the two types of error
exponents under limited rate is studied in the following way. First, the
problem is reduced to the problem of determining the reliability function of
channel codes designed for detection (in analogy to a similar result which
connects the reliability function of distributed lossless compression and
ordinary channel codes). Second, a single-letter random-coding bound based on a
hierarchical ensemble, as well as a single-letter expurgated bound, are derived
for the reliability of channel-detection codes. Both bounds are derived for a
system which employs the optimal detection rule. We conjecture that the
resulting random-coding bound is ensemble-tight, and consequently optimal
within the class of quantization-and-binning schemes
On the VC-Dimension of Binary Codes
We investigate the asymptotic rates of length- binary codes with
VC-dimension at most and minimum distance at least . Two upper
bounds are obtained, one as a simple corollary of a result by Haussler and the
other via a shortening approach combining Sauer-Shelah lemma and the linear
programming bound. Two lower bounds are given using Gilbert-Varshamov type
arguments over constant-weight and Markov-type sets
Expurgated Bounds for the Asymmetric Broadcast Channel
This work contains two main contributions concerning the expurgation of
hierarchical ensembles for the asymmetric broadcast channel. The first is an
analysis of the optimal maximum likelihood (ML) decoders for the weak and
strong user. Two different methods of code expurgation will be used, that will
provide two competing error exponents. The second is the derivation of
expurgated exponents under the generalized stochastic likelihood decoder (GLD).
We prove that the GLD exponents are at least as tight as the maximum between
the random coding error exponents derived in an earlier work by Averbuch and
Merhav (2017) and one of our ML-based expurgated exponents. By that, we
actually prove the existence of hierarchical codebooks that achieve the best of
the random coding exponent and the expurgated exponent simultaneously for both
users
Guessing with a Bit of Help
What is the value of a single bit to a guesser? We study this problem in a
setup where Alice wishes to guess an i.i.d. random vector, and can procure one
bit of information from Bob, who observes this vector through a memoryless
channel. We are interested in the guessing efficiency, which we define as the
best possible multiplicative reduction in Alice's guessing-moments obtainable
by observing Bob's bit. For the case of a uniform binary vector observed
through a binary symmetric channel, we provide two lower bounds on the guessing
efficiency by analyzing the performance of the Dictator and Majority functions,
and two upper bounds via maximum entropy and Fourier-analytic /
hypercontractivity arguments. We then extend our maximum entropy argument to
give a lower bound on the guessing efficiency for a general channel with a
binary uniform input, via the strong data-processing inequality constant of the
reverse channel. We compute this bound for the binary erasure channel, and
conjecture that Greedy Dictator functions achieve the guessing efficiency
Self-Predicting Boolean Functions
A Boolean function is said to be an optimal predictor for another Boolean
function , if it minimizes the probability that
among all functions, where is uniform over the Hamming cube and
is obtained from by independently flipping each coordinate with
probability . This paper is about self-predicting functions, which are
those that coincide with their optimal predictor
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