53,311 research outputs found
On the structure of optimal error-correcting codes
AbstractKabatyanskii and Panchenko asked whether two sets of size 10 consisting of binary 7-tuples exist, such that all 100 sums with one element from each set are distinct. This question is here answered in the negative by showing that the existence of such sets would imply the existence of a binary single-error-correcting code of length 9 and size 40 (which is unique) with a certain property, which such a code does not have
Error-Correcting Data Structures
We study data structures in the presence of adversarial noise. We want to
encode a given object in a succinct data structure that enables us to
efficiently answer specific queries about the object, even if the data
structure has been corrupted by a constant fraction of errors. This new model
is the common generalization of (static) data structures and locally decodable
error-correcting codes. The main issue is the tradeoff between the space used
by the data structure and the time (number of probes) needed to answer a query
about the encoded object. We prove a number of upper and lower bounds on
various natural error-correcting data structure problems. In particular, we
show that the optimal length of error-correcting data structures for the
Membership problem (where we want to store subsets of size s from a universe of
size n) is closely related to the optimal length of locally decodable codes for
s-bit strings.Comment: 15 pages LaTeX; an abridged version will appear in the Proceedings of
the STACS 2009 conferenc
It'll probably work out: improved list-decoding through random operations
In this work, we introduce a framework to study the effect of random
operations on the combinatorial list-decodability of a code. The operations we
consider correspond to row and column operations on the matrix obtained from
the code by stacking the codewords together as columns. This captures many
natural transformations on codes, such as puncturing, folding, and taking
subcodes; we show that many such operations can improve the list-decoding
properties of a code. There are two main points to this. First, our goal is to
advance our (combinatorial) understanding of list-decodability, by
understanding what structure (or lack thereof) is necessary to obtain it.
Second, we use our more general results to obtain a few interesting corollaries
for list decoding:
(1) We show the existence of binary codes that are combinatorially
list-decodable from fraction of errors with optimal rate
that can be encoded in linear time.
(2) We show that any code with relative distance, when randomly
folded, is combinatorially list-decodable fraction of errors with
high probability. This formalizes the intuition for why the folding operation
has been successful in obtaining codes with optimal list decoding parameters;
previously, all arguments used algebraic methods and worked only with specific
codes.
(3) We show that any code which is list-decodable with suboptimal list sizes
has many subcodes which have near-optimal list sizes, while retaining the error
correcting capabilities of the original code. This generalizes recent results
where subspace evasive sets have been used to reduce list sizes of codes that
achieve list decoding capacity
Improve the Usability of Polar Codes: Code Construction, Performance Enhancement and Configurable Hardware
Error-correcting codes (ECC) have been widely used for forward error correction (FEC) in modern communication systems to dramatically reduce the signal-to-noise ratio (SNR) needed to achieve a given bit error rate (BER). Newly invented polar codes have attracted much interest because of their capacity-achieving potential, efficient encoder and decoder implementation, and flexible architecture design space.This dissertation is aimed at improving the usability of polar codes by providing a practical code design method, new approaches to improve the performance of polar code, and a configurable hardware design that adapts to various specifications.
State-of-the-art polar codes are used to achieve extremely low error rates. In this work, high-performance FPGA is used in prototyping polar decoders to catch rare-case errors for error-correcting performance verification and error analysis. To discover the polarization characteristics and error patterns of polar codes, an FPGA emulation platform for belief-propagation (BP) decoding is built by a semi-automated construction flow. The FPGA-based emulation achieves significant speedup in large-scale experiments involving trillions of data frames. The platform is a key enabler of this work.
The frozen set selection of polar codes, known as bit selection, is critical to the error-correcting performance of polar codes. A simulation-based in-order bit selection method is developed to evaluate the error rate of each bit using Monte Carlo simulations. The frozen set is selected based on the bit reliability ranking. The resulting code construction exhibits up to 1 dB coding gain with respect to the conventional bit selection.
To further improve the coding gain of BP decoder for low-error-rate applications, the decoding error mechanisms are studied and analyzed, and the errors are classified based on their distinct signatures. Error detection is enabled by low-cost CRC concatenation, and post-processing algorithms targeting at each type of the error is designed to mitigate the vast majority of the decoding errors. The post-processor incurs only a small implementation overhead, but it provides more than an order of magnitude improvement of the error-correcting performance.
The regularity of the BP decoder structure offers many hardware architecture choices. Silicon area, power consumption, throughput and latency can be traded to reach the optimal design points for practical use cases. A comprehensive design space exploration reveals several practical architectures at different design points. The scalability of each architecture is also evaluated based on the implementation candidates.
For dynamic communication channels, such as wireless channels in the upcoming 5G applications, multiple codes of different lengths and code rates are needed to t varying channel conditions. To minimize implementation cost, a universal decoder architecture is proposed to support multiple codes through hardware reuse. A 40nm length- and rate-configurable polar decoder ASIC is demonstrated to fit various
communication environments and service requirements.PHDElectrical EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/140817/1/shuangsh_1.pd
Quantum Coding with Entanglement
Quantum error-correcting codes will be the ultimate enabler of a future
quantum computing or quantum communication device. This theory forms the
cornerstone of practical quantum information theory. We provide several
contributions to the theory of quantum error correction--mainly to the theory
of "entanglement-assisted" quantum error correction where the sender and
receiver share entanglement in the form of entangled bits (ebits) before
quantum communication begins. Our first contribution is an algorithm for
encoding and decoding an entanglement-assisted quantum block code. We then give
several formulas that determine the optimal number of ebits for an
entanglement-assisted code. The major contribution of this thesis is the
development of the theory of entanglement-assisted quantum convolutional
coding. A convolutional code is one that has memory and acts on an incoming
stream of qubits. We explicitly show how to encode and decode a stream of
information qubits with the help of ancilla qubits and ebits. Our
entanglement-assisted convolutional codes include those with a
Calderbank-Shor-Steane structure and those with a more general structure. We
then formulate convolutional protocols that correct errors in noisy
entanglement. Our final contribution is a unification of the theory of quantum
error correction--these unified convolutional codes exploit all of the known
resources for quantum redundancy.Comment: Ph.D. Thesis, University of Southern California, 2008, 193 pages, 2
tables, 12 figures, 9 limericks; Available at
http://digitallibrary.usc.edu/search/controller/view/usctheses-m1491.htm
Spin squeezed GKP codes for quantum error correction in atomic ensembles
GKP codes encode a qubit in displaced phase space combs of a
continuous-variable (CV) quantum system and are useful for correcting a variety
of high-weight photonic errors. Here we propose atomic ensemble analogues of
the single-mode CV GKP code by using the quantum central limit theorem to pull
back the phase space structure of a CV system to the compact phase space of a
quantum spin system. We study the optimal recovery performance of these codes
under error channels described by stochastic relaxation and isotropic ballistic
dephasing processes using the diversity combining approach for calculating
channel fidelity. We find that the spin GKP codes outperform other spin system
codes such as cat codes or binomial codes. Our spin GKP codes based on the
two-axis countertwisting interaction and superpositions of SU(2) coherent
states are direct spin analogues of the finite-energy CV GKP codes, whereas our
codes based on one-axis twisting do not yet have well-studied CV analogues. An
implementation of the spin GKP codes is proposed which uses the linear
combination of unitaries method, applicable to both the CV and spin GKP
settings. Finally, we discuss a fault-tolerant approximate gate set for quantum
computing with spin GKP-encoded qubits, obtained by translating gates from the
CV GKP setting using quantum central limit theorem.Comment: More details added to the previous versions with more figure
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