Decoding Multivariate Multiplicity Codes on Product Sets

by   Siddharth Bhandari, et al.

The multiplicity Schwartz-Zippel lemma bounds the total multiplicity of zeroes of a multivariate polynomial on a product set. This lemma motivates the multiplicity codes of Kopparty, Saraf and Yekhanin [J. ACM, 2014], who showed how to use this lemma to construct high-rate locally-decodable codes. However, the algorithmic results about these codes crucially rely on the fact that the polynomials are evaluated on a vector space and not an arbitrary product set. In this work, we show how to decode multivariate multiplicity codes of large multiplicities in polynomial time over finite product sets (over fields of large characteristic and zero characteristic). Previously such decoding algorithms were not known even for a positive fraction of errors. In contrast, our work goes all the way to the distance of the code and in particular exceeds both the unique decoding bound and the Johnson bound. For errors exceeding the Johnson bound, even combinatorial list-decodablity of these codes was not known. Our algorithm is an application of the classical polynomial method directly to the multivariate setting. In particular, we do not rely on a reduction from the multivariate to the univariate case as is typical of many of the existing results on decoding codes based on multivariate polynomials. However, a vanilla application of the polynomial method in the multivariate setting does not yield a polynomial upper bound on the list size. We obtain a polynomial bound on the list size by taking an alternative view of multivariate multiplicity codes. In this view, we glue all the partial derivatives of the same order together using a fresh set z of variables. We then apply the polynomial method by viewing this as a problem over the field 𝔽(z) of rational functions in z.


page 1

page 2

page 3

page 4


Improved decoding of Folded Reed-Solomon and Multiplicity Codes

In this work, we show new and improved error-correcting properties of fo...

Algorithmizing the Multiplicity Schwartz-Zippel Lemma

The multiplicity Schwartz-Zippel lemma asserts that over a field, a low-...

List Decoding of Insertion and Deletion Codes

Insertion and deletion (Insdel for short) errors are synchronization err...

Reed-Solomon Codes over Fields of Characteristic Zero

We study Reed-Solomon codes over arbitrary fields, inspired by several r...

Local decoding and testing of polynomials over grids

The well-known DeMillo-Lipton-Schwartz-Zippel lemma says that n-variate ...

On generalizing Descartes' rule of signs to hypersurfaces

We give partial generalizations of the classical Descartes' rule of sign...

Ideal-theoretic Explanation of Capacity-achieving Decoding

In this work, we present an abstract framework for some algebraic error-...

Please sign up or login with your details

Forgot password? Click here to reset