Faster Algorithms for Multivariate Interpolation with Multiplicities and Simultaneous Polynomial Approximations
Abstract
The interpolation step in the Guruswami-Sudan algorithm is a bivariate
interpolation problem with multiplicities commonly solved in the literature
using either structured linear algebra or basis reduction of polynomial
lattices. This problem has been extended to three or more variables; for this
generalization, all fast algorithms proposed so far rely on the lattice
approach. In this paper, we reduce this multivariate interpolation problem to a
problem of simultaneous polynomial approximations, which we solve using fast
structured linear algebra. This improves the best known complexity bounds for
the interpolation step of the list-decoding of Reed-Solomon codes,
Parvaresh-Vardy codes, and folded Reed-Solomon codes. In particular, for
Reed-Solomon list-decoding with re-encoding, our approach has complexity
$\mathcal{O}\tilde{~}(\ell^{\omega-1}m^2(n-k))$, where $\ell,m,n,k$ are the
list size, the multiplicity, the number of sample points and the dimension of
the code, and $\omega$ is the exponent of linear algebra; this accelerates the
previously fastest known algorithm by a factor of $\ell / m$.
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