We study the list-decodability of multiplicity codes. These codes, which are based on evaluations of high-degree polynomials and their derivatives, have rate approaching $1$ while simultaneously allowing for sublinear-time error-correction. In this paper, we show that multiplicity codes also admit powerful list-decoding and local list-decoding algorithms correcting a large fraction of errors. Stated simply, we give algorithms for recovering a polynomial given several evaluations of it and its derivatives, where possibly many of the given evaluations are incorrect.
Our first main result shows that univariate multiplicity codes over prime fields can be list-decoded upto the so called ``list-decoding capacity". Specifically, we show that univariate multiplicity codes of rate $R$ over prime fields can be list-decoded from $(1- R - \epsilon)$-fraction errors in polynomial time. This resembles the behavior of the Folded Reed-Solomon Codes of Guruswami and Rudra. The list-decoding algorithm is based on constructing a differential equation of which the desired codeword is a solution; this differential equation is then solved using a power-series approach (a variation of Hensel lifting) along with other algebraic ideas.
Our second main result is a list-decoding algorithm for decoding multivariate multiplicity codes upto their Johnson radius. The key ingredient of this algorithm is the construction of a special family of ``algebraically-repelling" curves passing through the points of ${\mathbb F}_q^m$; no moderate-degree multivariate polynomial over ${\mathbb F}_q^m$ can simultaneously vanish on all these curves. These curves enable us to reduce the decoding of multivariate multiplicity codes over ${\mathbb F}_q^m$ to several instances of decoding univariate multiplicity codes over the big field ${\mathbb F}_{q^m}$, for which such list-decoding algorithms are known.
As a corollary, we show how multivariate multiplicity codes of length $n$ and rate nearly $1$ can be locally list-decoded upto their Johnson radius in $O(n^{\epsilon})$ time.
