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Kim, John Y.; Kopparty, Swastik

doi:10.4086/toc.2017.v013a021

Cited by 8 publications

(10 citation statements)

References 9 publications

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“…Polynomial-method based algorithms. It would also be interesting to understand if the results in this paper and those in [KK17] can be proved via a more direct application of the polynomial method. Note that for large s, constant m, and fields of sufficiently large or zero characteristic, the algorithm in [BHKS21] is indeed directly based on a clean application of the polynomial method.…”

Section: Further Discussion and Open Problemsmentioning

confidence: 94%

“…We remark that these univariate decoding algorithms work for all choices of the set T ⊂ F. However, all known decoding algorithms for the multivariate setting (both Reed-Muller and larger order multiplicity codes), till recently, worked only when the underlying set T had a nice algebraic structure (eg., T = F) or when the degree d was very small (cf, the Reed-Muller list-decoding algorithm of Sudan [Sud97] and its multiplicity variant due to Guruswami & Sudan [GS99]). The s = 1 setting of the multivariate case (even the bivariate case), which corresponds to an algorithmic version of the classical SZ lemma (without multiplicities) was only recently resolved in a beautiful work of Kim and Kopparty [KK17]. Their algorithm does not seem to extend to the case of s > 1, and they mention this as one of the open problems.…”

Section: Main Result: Algorithmic Multivariate Sz Lemmamentioning

confidence: 99%

“…, x m−1 ] (as opposed to being viewed as an (m − 1)-variate polynomial with the coefficients coming from a univariate polynomial ring F[x m ].). This subtle difference also shows up in the way induction is done in our decoding algorithm for the multivariate case when compared to how Kim-Kopparty proceed in the decoding algorithm for multivariate Reed-Muller codes [KK17]. In fact, it is not clear to us that the results in this paper can be obtained if we set up the induction as in the work of Kim and Kopparty [KK17].…”

Section: Multivariate Schwartz Zippel Lemmamentioning

confidence: 93%

“…In Section 2, we give an overview of the main ideas in our algorithm and discuss how they relate to the algorithm of Kim and Kopparty [KK17] for decoding Reed-Muller codes. In Section 3, we discuss the necessary preliminaries including the definition and properties of multiplicity codes and describe a fine grained notion of distance for multiplicity codes that plays a crucial role in our proofs.…”

Section: Organizationmentioning

confidence: 99%

“…We discuss a proof of the standard Schwartz-Zippel lemma in this sections, that is slightly different from the standard textbook proof of the lemma, e.g. in [AB09] and the proof in Kim and Kopparty's work [KK17].…”

Section: Multivariate Schwartz Zippel Lemmamentioning

confidence: 99%

See 4 more Smart Citations

Algorithmizing the Multiplicity Schwartz-Zippel Lemma

Bhandari¹,

Harsha²,

Kumar³

et al. 2021

Preprint

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The multiplicity Schwartz-Zippel lemma asserts that over a field, a low-degree polynomial cannot vanish with high multiplicity very often on a sufficiently large product set. Since its discovery in a work of Dvir, Kopparty, Saraf and Sudan [DKSS13], the lemma has found numerous applications in both math and computer science; in particular, in the definition and properties of multiplicity codes by Kopparty, Saraf and Yekhanin [KSY14].In this work, we show how to algorithmize the multiplicity Schwartz-Zippel lemma for arbitrary product sets over any field. In other words, we give an efficient algorithm for unique decoding of multivariate multiplicity codes from half their minimum distance on arbitrary product sets over all fields. Previously, such an algorithm was known either when the underlying product set had a nice algebraic structure (for instance, was a subfield) [Kop15] or when the underlying field had large (or zero) characteristic, the multiplicity parameter was sufficiently large and the multiplicity code had distance bounded away from 1 [BHKS21]. In particular, even unique decoding of bivariate multiplicity codes with multiplicity two from half their minimum distance was not known over arbitrary product sets over any field.Our algorithm builds upon a result of Kim & Kopparty [KK17] who gave an algorithmic version of the Schwartz-Zippel lemma (without multiplicities) or equivalently, an efficient algorithm for unique decoding of Reed-Muller codes over arbitrary product sets. We introduce a refined notion of distance based on the multiplicity Schwartz-Zippel lemma and design a unique decoding algorithm for this distance measure. On the way, we give an alternate proof of Forney's classical generalized minimum distance decoder that might be of independent interest.

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Section: Further Discussion and Open Problemsmentioning

confidence: 94%

Section: Main Result: Algorithmic Multivariate Sz Lemmamentioning

confidence: 99%