253 research outputs found
Reed-Muller codes for random erasures and errors
This paper studies the parameters for which Reed-Muller (RM) codes over
can correct random erasures and random errors with high probability,
and in particular when can they achieve capacity for these two classical
channels. Necessarily, the paper also studies properties of evaluations of
multi-variate polynomials on random sets of inputs.
For erasures, we prove that RM codes achieve capacity both for very high rate
and very low rate regimes. For errors, we prove that RM codes achieve capacity
for very low rate regimes, and for very high rates, we show that they can
uniquely decode at about square root of the number of errors at capacity.
The proofs of these four results are based on different techniques, which we
find interesting in their own right. In particular, we study the following
questions about , the matrix whose rows are truth tables of all
monomials of degree in variables. What is the most (resp. least)
number of random columns in that define a submatrix having full column
rank (resp. full row rank) with high probability? We obtain tight bounds for
very small (resp. very large) degrees , which we use to show that RM codes
achieve capacity for erasures in these regimes.
Our decoding from random errors follows from the following novel reduction.
For every linear code of sufficiently high rate we construct a new code
, also of very high rate, such that for every subset of coordinates, if
can recover from erasures in , then can recover from errors in .
Specializing this to RM codes and using our results for erasures imply our
result on unique decoding of RM codes at high rate.
Finally, two of our capacity achieving results require tight bounds on the
weight distribution of RM codes. We obtain such bounds extending the recent
\cite{KLP} bounds from constant degree to linear degree polynomials
Efficiently decoding Reed-Muller codes from random errors
Reed-Muller codes encode an -variate polynomial of degree by
evaluating it on all points in . We denote this code by .
The minimal distance of is and so it cannot correct more
than half that number of errors in the worst case. For random errors one may
hope for a better result.
In this work we give an efficient algorithm (in the block length ) for
decoding random errors in Reed-Muller codes far beyond the minimal distance.
Specifically, for low rate codes (of degree ) we can correct a
random set of errors with high probability. For high rate codes
(of degree for ), we can correct roughly
errors.
More generally, for any integer , our algorithm can correct any error
pattern in for which the same erasure pattern can be corrected
in . The results above are obtained by applying recent results
of Abbe, Shpilka and Wigderson (STOC, 2015), Kumar and Pfister (2015) and
Kudekar et al. (2015) regarding the ability of Reed-Muller codes to correct
random erasures.
The algorithm is based on solving a carefully defined set of linear equations
and thus it is significantly different than other algorithms for decoding
Reed-Muller codes that are based on the recursive structure of the code. It can
be seen as a more explicit proof of a result of Abbe et al. that shows a
reduction from correcting erasures to correcting errors, and it also bares some
similarities with the famous Berlekamp-Welch algorithm for decoding
Reed-Solomon codes.Comment: 18 pages, 2 figure
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
List Decoding Tensor Products and Interleaved Codes
We design the first efficient algorithms and prove new combinatorial bounds
for list decoding tensor products of codes and interleaved codes. We show that
for {\em every} code, the ratio of its list decoding radius to its minimum
distance stays unchanged under the tensor product operation (rather than
squaring, as one might expect). This gives the first efficient list decoders
and new combinatorial bounds for some natural codes including multivariate
polynomials where the degree in each variable is bounded. We show that for {\em
every} code, its list decoding radius remains unchanged under -wise
interleaving for an integer . This generalizes a recent result of Dinur et
al \cite{DGKS}, who proved such a result for interleaved Hadamard codes
(equivalently, linear transformations). Using the notion of generalized Hamming
weights, we give better list size bounds for {\em both} tensoring and
interleaving of binary linear codes. By analyzing the weight distribution of
these codes, we reduce the task of bounding the list size to bounding the
number of close-by low-rank codewords. For decoding linear transformations,
using rank-reduction together with other ideas, we obtain list size bounds that
are tight over small fields.Comment: 32 page
Decoding Reed-Muller codes over product sets
We give a polynomial time algorithm to decode multivariate polynomial codes
of degree up to half their minimum distance, when the evaluation points are
an arbitrary product set , for every . Previously known
algorithms can achieve this only if the set has some very special algebraic
structure, or if the degree is significantly smaller than . We also
give a near-linear time randomized algorithm, which is based on tools from
list-decoding, to decode these codes from nearly half their minimum distance,
provided .
Our result gives an -dimensional generalization of the well known decoding
algorithms for Reed-Solomon codes, and can be viewed as giving an algorithmic
version of the Schwartz-Zippel lemma.Comment: 25 pages, 0 figure
Product Construction of Affine Codes
Binary matrix codes with restricted row and column weights are a desirable
method of coded modulation for power line communication. In this work, we
construct such matrix codes that are obtained as products of affine codes -
cosets of binary linear codes. Additionally, the constructions have the
property that they are systematic. Subsequently, we generalize our construction
to irregular product of affine codes, where the component codes are affine
codes of different rates.Comment: 13 pages, to appear in SIAM Journal on Discrete Mathematic
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