Abstract
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.
Highlights
Reed-Solomon codes and Reed-Muller codes are classical families of error-correcting codes which have found wide applicability within theoretical computer science
A typical setting of interest is where Fq is a large finite field, m, s are constants, and d = (1 − δ )sq for some δ ∈ (0, 1). (So δ becomes the minimum distance of this code.) The augmentation of the derivatives allows one to consider polynomials over Fq of degree larger than q, and this leads to much better tradeoffs in the rate and minimum distance of these codes, while retaining the good local decodability of Reed-Muller codes
We show that univariate multiplicity codes of rate R achieve “list-decoding capacity”; they can be list-decoded from a (1 − R − ε) fraction of errors in polynomial time with polynomial size lists
Summary
Reed-Solomon codes (which are codes based on univariate polynomials) and Reed-Muller codes (which are codes based on multivariate polynomials) are classical families of error-correcting codes which have found wide applicability within theoretical computer science. As a corollary of our second result, we show that m-variate m√ultiplicity codes of distance δ (which have rate ≈ (1 − δ )m) can be locally list-decoded from a (1 − 1 − δ − ε) fraction of errors in time O(nO(1/m)) This gives the first algorithm for local decoding of multivariate multiplicity codes. (The local decoding algorithm of [22] can at best decode from 1/4 the minimum distance.) This gives the best known tradeoff between rate, error tolerance, and query complexity of local decoding One goal of this line of research is to develop good algorithms to support error correction for multiplicity codes. Multiplicity codes are currently the only known codes which achieve rate arbitrarily close to 1 while supporting o(n)-time local decodability They could potentially be useful for real-life error correction.
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