8,981 research outputs found
Recent advances in coding theory for near error-free communications
Channel and source coding theories are discussed. The following subject areas are covered: large constraint length convolutional codes (the Galileo code); decoder design (the big Viterbi decoder); Voyager's and Galileo's data compression scheme; current research in data compression for images; neural networks for soft decoding; neural networks for source decoding; finite-state codes; and fractals for data compression
Convolutional Codes in Rank Metric with Application to Random Network Coding
Random network coding recently attracts attention as a technique to
disseminate information in a network. This paper considers a non-coherent
multi-shot network, where the unknown and time-variant network is used several
times. In order to create dependencies between the different shots, particular
convolutional codes in rank metric are used. These codes are so-called
(partial) unit memory ((P)UM) codes, i.e., convolutional codes with memory one.
First, distance measures for convolutional codes in rank metric are shown and
two constructions of (P)UM codes in rank metric based on the generator matrices
of maximum rank distance codes are presented. Second, an efficient
error-erasure decoding algorithm for these codes is presented. Its guaranteed
decoding radius is derived and its complexity is bounded. Finally, it is shown
how to apply these codes for error correction in random linear and affine
network coding.Comment: presented in part at Netcod 2012, submitted to IEEE Transactions on
Information Theor
Synchronization Strings: Codes for Insertions and Deletions Approaching the Singleton Bound
We introduce synchronization strings as a novel way of efficiently dealing
with synchronization errors, i.e., insertions and deletions. Synchronization
errors are strictly more general and much harder to deal with than commonly
considered half-errors, i.e., symbol corruptions and erasures. For every
, synchronization strings allow to index a sequence with an
size alphabet such that one can efficiently transform
synchronization errors into half-errors. This powerful new
technique has many applications. In this paper, we focus on designing insdel
codes, i.e., error correcting block codes (ECCs) for insertion deletion
channels.
While ECCs for both half-errors and synchronization errors have been
intensely studied, the later has largely resisted progress. Indeed, it took
until 1999 for the first insdel codes with constant rate, constant distance,
and constant alphabet size to be constructed by Schulman and Zuckerman. Insdel
codes for asymptotically large or small noise rates were given in 2016 by
Guruswami et al. but these codes are still polynomially far from the optimal
rate-distance tradeoff. This makes the understanding of insdel codes up to this
work equivalent to what was known for regular ECCs after Forney introduced
concatenated codes in his doctoral thesis 50 years ago.
A direct application of our synchronization strings based indexing method
gives a simple black-box construction which transforms any ECC into an equally
efficient insdel code with a slightly larger alphabet size. This instantly
transfers much of the highly developed understanding for regular ECCs over
large constant alphabets into the realm of insdel codes. Most notably, we
obtain efficient insdel codes which get arbitrarily close to the optimal
rate-distance tradeoff given by the Singleton bound for the complete noise
spectrum
Subspace Evasive Sets
In this work we describe an explicit, simple, construction of large subsets
of F^n, where F is a finite field, that have small intersection with every
k-dimensional affine subspace. Interest in the explicit construction of such
sets, termed subspace-evasive sets, started in the work of Pudlak and Rodl
(2004) who showed how such constructions over the binary field can be used to
construct explicit Ramsey graphs. More recently, Guruswami (2011) showed that,
over large finite fields (of size polynomial in n), subspace evasive sets can
be used to obtain explicit list-decodable codes with optimal rate and constant
list-size. In this work we construct subspace evasive sets over large fields
and use them to reduce the list size of folded Reed-Solomon codes form poly(n)
to a constant.Comment: 16 page
Optimal rate list decoding via derivative codes
The classical family of Reed-Solomon codes over a field \F_q
consist of the evaluations of polynomials f \in \F_q[X] of degree at
distinct field elements. In this work, we consider a closely related family
of codes, called (order ) {\em derivative codes} and defined over fields of
large characteristic, which consist of the evaluations of as well as its
first formal derivatives at distinct field elements. For large enough
, we show that these codes can be list-decoded in polynomial time from an
error fraction approaching , where is the rate of the code.
This gives an alternate construction to folded Reed-Solomon codes for achieving
the optimal trade-off between rate and list error-correction radius. Our
decoding algorithm is linear-algebraic, and involves solving a linear system to
interpolate a multivariate polynomial, and then solving another structured
linear system to retrieve the list of candidate polynomials . The algorithm
for derivative codes offers some advantages compared to a similar one for
folded Reed-Solomon codes in terms of efficient unique decoding in the presence
of side information.Comment: 11 page
Fast Decoders for Topological Quantum Codes
We present a family of algorithms, combining real-space renormalization
methods and belief propagation, to estimate the free energy of a topologically
ordered system in the presence of defects. Such an algorithm is needed to
preserve the quantum information stored in the ground space of a topologically
ordered system and to decode topological error-correcting codes. For a system
of linear size L, our algorithm runs in time log L compared to L^6 needed for
the minimum-weight perfect matching algorithm previously used in this context
and achieves a higher depolarizing error threshold.Comment: 4 pages, 4 figure
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