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Volume 11 (2015) Article 5 pp. 149-182
List-Decoding Multiplicity Codes
Received: February 4, 2013
Revised: April 1, 2015
Published: May 29, 2015
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Keywords: error-correcting codes, list-decoding, polynomials, differential equations
ACM Classification: E.4, G.1.7
AMS Classification: 11T71, 94B35, 68W30

Abstract: [Plain Text Version]

$ \newcommand{\F}{{\mathbb F}} $

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 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 $(1- R - \epsilon)$ fraction of errors in polynomial time (for constant $R, \epsilon$). This resembles the behavior of the “Folded Reed-Solomon Codes” of Guruswami and Rudra (Trans. Info. Theory 2008). 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 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 $\F_q^m$; no moderate-degree multivariate polynomial over $\F_q^m$ can simultaneously vanish on all these curves. These curves enable us to reduce the decoding of multivariate multiplicity codes over $\F_q^m$ to several instances of decoding univariate multiplicity codes over the big field $\F_{q^m}$, 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 up to their Johnson radius in $O(n^{\epsilon})$ time.

A version of this paper was posted online as an Electronic Colloq. on Computational Complexity (ECCC) Tech. Report.