27,482 research outputs found
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
Performance of polar codes for quantum and private classical communication
We analyze the practical performance of quantum polar codes, by computing
rigorous bounds on block error probability and by numerically simulating them.
We evaluate our bounds for quantum erasure channels with coding block lengths
between 2^10 and 2^20, and we report the results of simulations for quantum
erasure channels, quantum depolarizing channels, and "BB84" channels with
coding block lengths up to N = 1024. For quantum erasure channels, we observe
that high quantum data rates can be achieved for block error rates less than
10^(-4) and that somewhat lower quantum data rates can be achieved for quantum
depolarizing and BB84 channels. Our results here also serve as bounds for and
simulations of private classical data transmission over these channels,
essentially due to Renes' duality bounds for privacy amplification and
classical data transmission of complementary observables. Future work might be
able to improve upon our numerical results for quantum depolarizing and BB84
channels by employing a polar coding rule other than the heuristic used here.Comment: 8 pages, 6 figures, submission to the 50th Annual Allerton Conference
on Communication, Control, and Computing 201
On privacy amplification, lossy compression, and their duality to channel coding
We examine the task of privacy amplification from information-theoretic and
coding-theoretic points of view. In the former, we give a one-shot
characterization of the optimal rate of privacy amplification against classical
adversaries in terms of the optimal type-II error in asymmetric hypothesis
testing. This formulation can be easily computed to give finite-blocklength
bounds and turns out to be equivalent to smooth min-entropy bounds by Renner
and Wolf [Asiacrypt 2005] and Watanabe and Hayashi [ISIT 2013], as well as a
bound in terms of the divergence by Yang, Schaefer, and Poor
[arXiv:1706.03866 [cs.IT]]. In the latter, we show that protocols for privacy
amplification based on linear codes can be easily repurposed for channel
simulation. Combined with known relations between channel simulation and lossy
source coding, this implies that privacy amplification can be understood as a
basic primitive for both channel simulation and lossy compression. Applied to
symmetric channels or lossy compression settings, our construction leads to
proto- cols of optimal rate in the asymptotic i.i.d. limit. Finally, appealing
to the notion of channel duality recently detailed by us in [IEEE Trans. Info.
Theory 64, 577 (2018)], we show that linear error-correcting codes for
symmetric channels with quantum output can be transformed into linear lossy
source coding schemes for classical variables arising from the dual channel.
This explains a "curious duality" in these problems for the (self-dual) erasure
channel observed by Martinian and Yedidia [Allerton 2003; arXiv:cs/0408008] and
partly anticipates recent results on optimal lossy compression by polar and
low-density generator matrix codes.Comment: v3: updated to include equivalence of the converse bound with smooth
entropy formulations. v2: updated to include comparison with the one-shot
bounds of arXiv:1706.03866. v1: 11 pages, 4 figure
A Minimax Converse for Quantum Channel Coding
We prove a one-shot "minimax" converse bound for quantum channel coding
assisted by positive partial transpose channels between sender and receiver.
The bound is similar in spirit to the converse by Polyanskiy, Poor, and Verdu
[IEEE Trans. Info. Theory 56, 2307-2359 (2010)] for classical channel coding,
and also enjoys the saddle point property enabling the order of optimizations
to be interchanged. Equivalently, the bound can be formulated as a semidefinite
program satisfying strong duality. The convex nature of the bound implies
channel symmetries can substantially simplify the optimization, enabling us to
explicitly compute the finite blocklength behavior for several simple qubit
channels. In particular, we find that finite blocklength converse statements
for the classical erasure channel apply to the assisted quantum erasure
channel, while bounds for the classical binary symmetric channel apply to both
the assisted dephasing and depolarizing channels. This implies that these qubit
channels inherit statements regarding the asymptotic limit of large
blocklength, such as the strong converse or second-order converse rates, from
their classical counterparts. Moreover, for the dephasing channel, the finite
blocklength bounds are as tight as those for the classical binary symmetric
channel, since coding for classical phase errors yields equivalently-performing
unassisted quantum codes.Comment: merged with arXiv:1504.04617 version 1 ; see version
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