231 research outputs found
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
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
Reed-Solomon list decoding from a system-theoretic perspective
In this paper, the Sudan-Guruswami approach to list decoding of Reed-Solomon (RS) codes is cast in a system-theoretic framework. With the data, a set of trajectories or time series is associated which is then modeled as a so-called behavior. In this way, a connection is made with the behavioral approach to system theory. It is shown how a polynomial representation of the modeling behavior gives rise to the bivariate interpolating polynomials of the Sudan-Guruswami approach. The concept of "weighted row reduced" is introduced and used to achieve minimality. Two decoding methods are derived and a parametrization of all bivariate interpolating polynomials is given
A behavioral framework for Reed-Solomon decoding through multiplicative bivariate interpolation
A behavioral setting is provided for decoding of Reed-Solomon codes. More specifically Sudan's approach based on bivariate polynomial interpolation is cast into a behavioral setting. The interpolation problem is translated into a linear behavioral modelling problem. The notion of weighted row reduced representation turns out to be crucial
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