164 research outputs found
Fast Computation of Minimal Interpolation Bases in Popov Form for Arbitrary Shifts
We compute minimal bases of solutions for a general interpolation problem,
which encompasses Hermite-Pad\'e approximation and constrained multivariate
interpolation, and has applications in coding theory and security.
This problem asks to find univariate polynomial relations between vectors
of size ; these relations should have small degree with respect to an
input degree shift. For an arbitrary shift, we propose an algorithm for the
computation of an interpolation basis in shifted Popov normal form with a cost
of field operations, where
is the exponent of matrix multiplication and the notation
indicates that logarithmic terms are omitted.
Earlier works, in the case of Hermite-Pad\'e approximation and in the general
interpolation case, compute non-normalized bases. Since for arbitrary shifts
such bases may have size , the cost bound
was feasible only with restrictive
assumptions on the shift that ensure small output sizes. The question of
handling arbitrary shifts with the same complexity bound was left open.
To obtain the target cost for any shift, we strengthen the properties of the
output bases, and of those obtained during the course of the algorithm: all the
bases are computed in shifted Popov form, whose size is always . Then, we design a divide-and-conquer scheme. We recursively reduce
the initial interpolation problem to sub-problems with more convenient shifts
by first computing information on the degrees of the intermediate bases.Comment: 8 pages, sig-alternate class, 4 figures (problems and algorithms
Sub-quadratic Decoding of One-point Hermitian Codes
We present the first two sub-quadratic complexity decoding algorithms for
one-point Hermitian codes. The first is based on a fast realisation of the
Guruswami-Sudan algorithm by using state-of-the-art algorithms from computer
algebra for polynomial-ring matrix minimisation. The second is a Power decoding
algorithm: an extension of classical key equation decoding which gives a
probabilistic decoding algorithm up to the Sudan radius. We show how the
resulting key equations can be solved by the same methods from computer
algebra, yielding similar asymptotic complexities.Comment: New version includes simulation results, improves some complexity
results, as well as a number of reviewer corrections. 20 page
Application of Computer Algebra in List Decoding
The amount of data that we use in everyday life (social media, stock analysis, satellite communication etc.) are increasing day by day. As a result, the amount of data needs to be traverse through electronic media as well as to store are rapidly growing and there exist several environmental effects that can damage these important data during travelling or while in storage devices. To recover correct information from noisy data, we do use error correcting codes. The most challenging work in this area is to have a decoding algorithm that can decode the code quite fast, in addition with the existence of the code that can tolerate highest amount of noise, so that we can have it in practice.
List decoding is an active research area for last two decades. This research popularise in coding theory after the breakthrough work by Madhu Sudan where he used list decoding technique to correct errors that exceeds half the minimum distance of Reed Solomon codes. Towards the direction of code development that can reach theoretical limit of error correction, Guruswami-Rudra introduced folded Reed Solomon codes that reached at To decode this codes, one has to first interpolate a multivariate polynomial first and then have to factor out all possible roots. The difficulties that lies here are efficient interpolation, dealing with multiplicities smartly and efficient factoring. This thesis deals with all these cases in order to have folded Reed Solomon codes in practice
Computing minimal interpolation bases
International audienceWe consider the problem of computing univariate polynomial matrices over afield that represent minimal solution bases for a general interpolationproblem, some forms of which are the vector M-Pad\'e approximation problem in[Van Barel and Bultheel, Numerical Algorithms 3, 1992] and the rationalinterpolation problem in [Beckermann and Labahn, SIAM J. Matrix Anal. Appl. 22,2000]. Particular instances of this problem include the bivariate interpolationsteps of Guruswami-Sudan hard-decision and K\"otter-Vardy soft-decisiondecodings of Reed-Solomon codes, the multivariate interpolation step oflist-decoding of folded Reed-Solomon codes, and Hermite-Pad\'e approximation. In the mentioned references, the problem is solved using iterative algorithmsbased on recurrence relations. Here, we discuss a fast, divide-and-conquerversion of this recurrence, taking advantage of fast matrix computations overthe scalars and over the polynomials. This new algorithm is deterministic, andfor computing shifted minimal bases of relations between vectors of size it uses field operations, where is the exponent of matrix multiplication, and is the sum of theentries of the input shift , with . This complexity boundimproves in particular on earlier algorithms in the case of bivariateinterpolation for soft decoding, while matching fastest existing algorithms forsimultaneous Hermite-Pad\'e approximation
Multi-Trial GuruswamiâSudan Decoding for Generalised ReedâSolomon Codes
An iterated refinement procedure for the Guruswami--Sudan list decoding
algorithm for Generalised Reed--Solomon codes based on Alekhnovich's module
minimisation is proposed. The method is parametrisable and allows variants of
the usual list decoding approach. In particular, finding the list of
\emph{closest} codewords within an intermediate radius can be performed with
improved average-case complexity while retaining the worst-case complexity.Comment: WCC 2013 International Workshop on Coding and Cryptography (2013
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