11,328 research outputs found
Nonasymptotic coding-rate bounds for binary erasure channels with feedback
We present nonasymptotic achievability and converse bounds on the maximum coding rate (for a fixed average error probability and a fixed average blocklength) of variable-length full-feedback (VLF) and variable-length stop-feedback (VLSF) codes operating over a binary erasure channel (BEC). For the VLF setup, the achievability bound relies on a scheme that maps each message onto a variable-length Huffman codeword and then repeats each bit of the codeword until it is received correctly. The converse bound is inspired by the meta-converse framework by Polyanskiy, Poor, and VerduÌ (2010) and relies on binary sequential hypothesis testing. For the case of zero error probability, our achievability and converse bounds match. For the VLSF case, we provide achievability bounds that exploit the following feature of BEC: the decoder can assess the correctness of its estimate by verifying whether the chosen codeword is the only one that is compatible with the erasure pattern. One of these bounds is obtained by analyzing the performance of a variable-length extension of random linear fountain codes. The gap between the VLSF achievability and the VLF converse bound, when number of messages is small, is significant: 23% for 8 messages on a BEC with erasure probability 0.5. The absence of a tight VLSF converse bound does not allow us to assess whether this gap is fundamental
A Tight Upper Bound for the Third-Order Asymptotics for Most Discrete Memoryless Channels
This paper shows that the logarithm of the epsilon-error capacity (average
error probability) for n uses of a discrete memoryless channel is upper bounded
by the normal approximation plus a third-order term that does not exceed 1/2
log n + O(1) if the epsilon-dispersion of the channel is positive. This matches
a lower bound by Y. Polyanskiy (2010) for discrete memoryless channels with
positive reverse dispersion. If the epsilon-dispersion vanishes, the logarithm
of the epsilon-error capacity is upper bounded by the n times the capacity plus
a constant term except for a small class of DMCs and epsilon >= 1/2.Comment: published versio
Multiple Quantum Hypothesis Testing Expressions and Classical-Quantum Channel Converse Bounds
Alternative exact expressions are derived for the minimum error probability
of a hypothesis test discriminating among quantum states. The first
expression corresponds to the error probability of a binary hypothesis test
with certain parameters; the second involves the optimization of a given
information-spectrum measure. Particularized in the classical-quantum channel
coding setting, this characterization implies the tightness of two existing
converse bounds; one derived by Matthews and Wehner using hypothesis-testing,
and one obtained by Hayashi and Nagaoka via an information-spectrum approach.Comment: Presented at the 2016 IEEE International Symposium on Information
Theory, July 10-15, 2016, Barcelona, Spai
Algorithmic Aspects of Optimal Channel Coding
A central question in information theory is to determine the maximum success
probability that can be achieved in sending a fixed number of messages over a
noisy channel. This was first studied in the pioneering work of Shannon who
established a simple expression characterizing this quantity in the limit of
multiple independent uses of the channel. Here we consider the general setting
with only one use of the channel. We observe that the maximum success
probability can be expressed as the maximum value of a submodular function.
Using this connection, we establish the following results:
1. There is a simple greedy polynomial-time algorithm that computes a code
achieving a (1-1/e)-approximation of the maximum success probability. Moreover,
for this problem it is NP-hard to obtain an approximation ratio strictly better
than (1-1/e).
2. Shared quantum entanglement between the sender and the receiver can
increase the success probability by a factor of at most 1/(1-1/e). In addition,
this factor is tight if one allows an arbitrary non-signaling box between the
sender and the receiver.
3. We give tight bounds on the one-shot performance of the meta-converse of
Polyanskiy-Poor-Verdu.Comment: v2: 16 pages. Added alternate proof of main result with random codin
On converse bounds for classical communication over quantum channels
We explore several new converse bounds for classical communication over
quantum channels in both the one-shot and asymptotic regimes. First, we show
that the Matthews-Wehner meta-converse bound for entanglement-assisted
classical communication can be achieved by activated, no-signalling assisted
codes, suitably generalizing a result for classical channels. Second, we derive
a new efficiently computable meta-converse on the amount of classical
information unassisted codes can transmit over a single use of a quantum
channel. As applications, we provide a finite resource analysis of classical
communication over quantum erasure channels, including the second-order and
moderate deviation asymptotics. Third, we explore the asymptotic analogue of
our new meta-converse, the -information of the channel. We show that
its regularization is an upper bound on the classical capacity, which is
generally tighter than the entanglement-assisted capacity and other known
efficiently computable strong converse bounds. For covariant channels we show
that the -information is a strong converse bound.Comment: v3: published version; v2: 18 pages, presentation and results
improve
Improved Finite Blocklength Converses for Slepian-Wolf Coding via Linear Programming
A new finite blocklength converse for the Slepian- Wolf coding problem is
presented which significantly improves on the best known converse for this
problem, due to Miyake and Kanaya [2]. To obtain this converse, an extension of
the linear programming (LP) based framework for finite blocklength point-
to-point coding problems from [3] is employed. However, a direct application of
this framework demands a complicated analysis for the Slepian-Wolf problem. An
analytically simpler approach is presented wherein LP-based finite blocklength
converses for this problem are synthesized from point-to-point lossless source
coding problems with perfect side-information at the decoder. New finite
blocklength metaconverses for these point-to-point problems are derived by
employing the LP-based framework, and the new converse for Slepian-Wolf coding
is obtained by an appropriate combination of these converses.Comment: under review with the IEEE Transactions on Information Theor
Coding in the Finite-Blocklength Regime: Bounds based on Laplace Integrals and their Asymptotic Approximations
In this paper we provide new compact integral expressions and associated
simple asymptotic approximations for converse and achievability bounds in the
finite blocklength regime. The chosen converse and random coding union bounds
were taken from the recent work of Polyanskyi-Poor-Verdu, and are investigated
under parallel AWGN channels, the AWGN channels, the BI-AWGN channel, and the
BSC. The technique we use, which is a generalization of some recent results
available from the literature, is to map the probabilities of interest into a
Laplace integral, and then solve (or approximate) the integral by use of a
steepest descent technique. The proposed results are particularly useful for
short packet lengths, where the normal approximation may provide unreliable
results.Comment: 29 pages, 10 figures. Submitted to IEEE Trans. on Information Theory.
Matlab code available from http://dgt.dei.unipd.it section Download->Finite
Blocklength Regim
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