List-Decoding Multiplicity Codes.
Swastik Kopparty · 2012
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 and local algorithms that work even in the presence of a large error fraction. In other words, 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 fields of prime order can be list-decoded up to the so-called list-decoding capacity. Specifically, we show that univariate multiplicity codes of rate R over fields of prime order can be list-decoded from a (1 R e) fraction of errors in polynomial time (for constant R;e). This resembles the behavior of the Folded Reed-Solomon Codes of Guruswami and Rudra (Trans. Info. Theory 2008). The 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 algorithm for decoding multivariate multiplicity codes up to 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 F m ; no moderate-degree multivariate polynomial over F m can simultaneously vanish on all these A version of this paper was posted online as an Electronic Colloq. on Computational Complexity Tech. Report (20). Supported in part by a Sloan Fellowship and NSF grant CCF-1253886.