1,866 research outputs found
Optimal Threshold-Based Multi-Trial Error/Erasure Decoding with the Guruswami-Sudan Algorithm
Traditionally, multi-trial error/erasure decoding of Reed-Solomon (RS) codes
is based on Bounded Minimum Distance (BMD) decoders with an erasure option.
Such decoders have error/erasure tradeoff factor L=2, which means that an error
is twice as expensive as an erasure in terms of the code's minimum distance.
The Guruswami-Sudan (GS) list decoder can be considered as state of the art in
algebraic decoding of RS codes. Besides an erasure option, it allows to adjust
L to values in the range 1<L<=2. Based on previous work, we provide formulae
which allow to optimally (in terms of residual codeword error probability)
exploit the erasure option of decoders with arbitrary L, if the decoder can be
used z>=1 times. We show that BMD decoders with z_BMD decoding trials can
result in lower residual codeword error probability than GS decoders with z_GS
trials, if z_BMD is only slightly larger than z_GS. This is of practical
interest since BMD decoders generally have lower computational complexity than
GS decoders.Comment: Accepted for the 2011 IEEE International Symposium on Information
Theory, St. Petersburg, Russia, July 31 - August 05, 2011. 5 pages, 2 figure
On Multiple Decoding Attempts for Reed-Solomon Codes: A Rate-Distortion Approach
One popular approach to soft-decision decoding of Reed-Solomon (RS) codes is
based on using multiple trials of a simple RS decoding algorithm in combination
with erasing or flipping a set of symbols or bits in each trial. This paper
presents a framework based on rate-distortion (RD) theory to analyze these
multiple-decoding algorithms. By defining an appropriate distortion measure
between an error pattern and an erasure pattern, the successful decoding
condition, for a single errors-and-erasures decoding trial, becomes equivalent
to distortion being less than a fixed threshold. Finding the best set of
erasure patterns also turns into a covering problem which can be solved
asymptotically by rate-distortion theory. Thus, the proposed approach can be
used to understand the asymptotic performance-versus-complexity trade-off of
multiple errors-and-erasures decoding of RS codes.
This initial result is also extended a few directions. The rate-distortion
exponent (RDE) is computed to give more precise results for moderate
blocklengths. Multiple trials of algebraic soft-decision (ASD) decoding are
analyzed using this framework. Analytical and numerical computations of the RD
and RDE functions are also presented. Finally, simulation results show that
sets of erasure patterns designed using the proposed methods outperform other
algorithms with the same number of decoding trials.Comment: to appear in the IEEE Transactions on Information Theory (Special
Issue on Facets of Coding Theory: from Algorithms to Networks
Optimal Thresholds for GMD Decoding with (L+1)/L-extended Bounded Distance Decoders
We investigate threshold-based multi-trial decoding of concatenated codes
with an inner Maximum-Likelihood decoder and an outer error/erasure
(L+1)/L-extended Bounded Distance decoder, i.e. a decoder which corrects e
errors and t erasures if e(L+1)/L + t <= d - 1, where d is the minimum distance
of the outer code and L is a positive integer. This is a generalization of
Forney's GMD decoding, which was considered only for L = 1, i.e. outer Bounded
Minimum Distance decoding. One important example for (L+1)/L-extended Bounded
Distance decoders is decoding of L-Interleaved Reed-Solomon codes. Our main
contribution is a threshold location formula, which allows to optimally erase
unreliable inner decoding results, for a given number of decoding trials and
parameter L. Thereby, the term optimal means that the residual codeword error
probability of the concatenated code is minimized. We give an estimation of
this probability for any number of decoding trials.Comment: Accepted for the 2010 IEEE International Symposium on Information
Theory, Austin, TX, USA, June 13 - 18, 2010. 5 pages, 2 figure
A Rate-Distortion Exponent Approach to Multiple Decoding Attempts for Reed-Solomon Codes
Algorithms based on multiple decoding attempts of Reed-Solomon (RS) codes
have recently attracted new attention. Choosing decoding candidates based on
rate-distortion (R-D) theory, as proposed previously by the authors, currently
provides the best performance-versus-complexity trade-off. In this paper, an
analysis based on the rate-distortion exponent (RDE) is used to directly
minimize the exponential decay rate of the error probability. This enables
rigorous bounds on the error probability for finite-length RS codes and leads
to modest performance gains. As a byproduct, a numerical method is derived that
computes the rate-distortion exponent for independent non-identical sources.
Analytical results are given for errors/erasures decoding.Comment: accepted for presentation at 2010 IEEE International Symposium on
Information Theory (ISIT 2010), Austin TX, US
A New Chase-type Soft-decision Decoding Algorithm for Reed-Solomon Codes
This paper addresses three relevant issues arising in designing Chase-type
algorithms for Reed-Solomon codes: 1) how to choose the set of testing
patterns; 2) given the set of testing patterns, what is the optimal testing
order in the sense that the most-likely codeword is expected to appear earlier;
and 3) how to identify the most-likely codeword. A new Chase-type soft-decision
decoding algorithm is proposed, referred to as tree-based Chase-type algorithm.
The proposed algorithm takes the set of all vectors as the set of testing
patterns, and hence definitely delivers the most-likely codeword provided that
the computational resources are allowed. All the testing patterns are arranged
in an ordered rooted tree according to the likelihood bounds of the possibly
generated codewords. While performing the algorithm, the ordered rooted tree is
constructed progressively by adding at most two leafs at each trial. The
ordered tree naturally induces a sufficient condition for the most-likely
codeword. That is, whenever the proposed algorithm exits before a preset
maximum number of trials is reached, the output codeword must be the
most-likely one. When the proposed algorithm is combined with Guruswami-Sudan
(GS) algorithm, each trial can be implement in an extremely simple way by
removing one old point and interpolating one new point. Simulation results show
that the proposed algorithm performs better than the recently proposed
Chase-type algorithm by Bellorado et al with less trials given that the maximum
number of trials is the same. Also proposed are simulation-based performance
bounds on the MLD algorithm, which are utilized to illustrate the
near-optimality of the proposed algorithm in the high SNR region. In addition,
the proposed algorithm admits decoding with a likelihood threshold, that
searches the most-likely codeword within an Euclidean sphere rather than a
Hamming sphere
Communication Efficient Secret Sharing
A secret sharing scheme is a method to store information securely and
reliably. Particularly, in a threshold secret sharing scheme, a secret is
encoded into shares, such that any set of at least shares suffice to
decode the secret, and any set of at most shares reveal no
information about the secret. Assuming that each party holds a share and a user
wishes to decode the secret by receiving information from a set of parties; the
question we study is how to minimize the amount of communication between the
user and the parties. We show that the necessary amount of communication,
termed "decoding bandwidth", decreases as the number of parties that
participate in decoding increases. We prove a tight lower bound on the decoding
bandwidth, and construct secret sharing schemes achieving the bound.
Particularly, we design a scheme that achieves the optimal decoding bandwidth
when parties participate in decoding, universally for all . The scheme is based on Shamir's secret sharing scheme and preserves its
simplicity and efficiency. In addition, we consider secure distributed storage
where the proposed communication efficient secret sharing schemes further
improve disk access complexity during decoding.Comment: submitted to the IEEE Transactions on Information Theory. New
references and a new construction adde
Optimal Iris Fuzzy Sketches
Fuzzy sketches, introduced as a link between biometry and cryptography, are a
way of handling biometric data matching as an error correction issue. We focus
here on iris biometrics and look for the best error-correcting code in that
respect. We show that two-dimensional iterative min-sum decoding leads to
results near the theoretical limits. In particular, we experiment our
techniques on the Iris Challenge Evaluation (ICE) database and validate our
findings.Comment: 9 pages. Submitted to the IEEE Conference on Biometrics: Theory,
Applications and Systems, 2007 Washington D
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