288 research outputs found
List and Probabilistic Unique Decoding of Folded Subspace Codes
A new class of folded subspace codes for noncoherent network coding is
presented. The codes can correct insertions and deletions beyond the unique
decoding radius for any code rate . An efficient interpolation-based
decoding algorithm for this code construction is given which allows to correct
insertions and deletions up to the normalized radius ,
where is the folding parameter and is a decoding parameter. The
algorithm serves as a list decoder or as a probabilistic unique decoder that
outputs a unique solution with high probability. An upper bound on the average
list size of (folded) subspace codes and on the decoding failure probability is
derived. A major benefit of the decoding scheme is that it enables
probabilistic unique decoding up to the list decoding radius.Comment: 6 pages, 1 figure, accepted for ISIT 201
Linear-algebraic list decoding of folded Reed-Solomon codes
Folded Reed-Solomon codes are an explicit family of codes that achieve the
optimal trade-off between rate and error-correction capability: specifically,
for any \eps > 0, the author and Rudra (2006,08) presented an n^{O(1/\eps)}
time algorithm to list decode appropriate folded RS codes of rate from a
fraction 1-R-\eps of errors. The algorithm is based on multivariate
polynomial interpolation and root-finding over extension fields. It was noted
by Vadhan that interpolating a linear polynomial suffices if one settles for a
smaller decoding radius (but still enough for a statement of the above form).
Here we give a simple linear-algebra based analysis of this variant that
eliminates the need for the computationally expensive root-finding step over
extension fields (and indeed any mention of extension fields). The entire list
decoding algorithm is linear-algebraic, solving one linear system for the
interpolation step, and another linear system to find a small subspace of
candidate solutions. Except for the step of pruning this subspace, the
algorithm can be implemented to run in {\em quadratic} time. The theoretical
drawback of folded RS codes are that both the decoding complexity and proven
worst-case list-size bound are n^{\Omega(1/\eps)}. By combining the above
idea with a pseudorandom subset of all polynomials as messages, we get a Monte
Carlo construction achieving a list size bound of O(1/\eps^2) which is quite
close to the existential O(1/\eps) bound (however, the decoding complexity
remains n^{\Omega(1/\eps)}). Our work highlights that constructing an
explicit {\em subspace-evasive} subset that has small intersection with
low-dimensional subspaces could lead to explicit codes with better
list-decoding guarantees.Comment: 16 pages. Extended abstract in Proc. of IEEE Conference on
Computational Complexity (CCC), 201
List and Unique Error-Erasure Decoding of Interleaved Gabidulin Codes with Interpolation Techniques
A new interpolation-based decoding principle for interleaved Gabidulin codes
is presented. The approach consists of two steps: First, a multi-variate
linearized polynomial is constructed which interpolates the coefficients of the
received word and second, the roots of this polynomial have to be found. Due to
the specific structure of the interpolation polynomial, both steps
(interpolation and root-finding) can be accomplished by solving a linear system
of equations. This decoding principle can be applied as a list decoding
algorithm (where the list size is not necessarily bounded polynomially) as well
as an efficient probabilistic unique decoding algorithm. For the unique
decoder, we show a connection to known unique decoding approaches and give an
upper bound on the failure probability. Finally, we generalize our approach to
incorporate not only errors, but also row and column erasures.Comment: accepted for Designs, Codes and Cryptography; presented in part at
WCC 2013, Bergen, Norwa
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
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
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