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Electronic Colloquium on Computational Complexity

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Reports tagged with multiplicity codes:
TR10-148 | 23rd September 2010
Swastik Kopparty, Shubhangi Saraf, Sergey Yekhanin

High-rate codes with sublinear-time decoding

Locally decodable codes are error-correcting codes that admit efficient decoding algorithms; any bit of the original message can be recovered by looking at only a small number of locations of a corrupted codeword. The tradeoff between the rate of a code and the locality/efficiency of its decoding algorithms has been ... more >>>

TR12-044 | 22nd April 2012
Swastik Kopparty

List-Decoding Multiplicity Codes

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 correcting a large fraction ... more >>>

TR18-091 | 4th May 2018
Swastik Kopparty, Noga Ron-Zewi, Shubhangi Saraf, Mary Wootters

Improved decoding of Folded Reed-Solomon and Multiplicity Codes

Revisions: 1

In this work, we show new and improved error-correcting properties of folded Reed-Solomon codes and multiplicity codes. Both of these families of codes are based on polynomials over finite fields, and both have been the sources of recent advances in coding theory. Folded Reed-Solomon codes were the first explicit constructions ... more >>>

TR21-025 | 15th February 2021
Sivakanth Gopi, Venkatesan Guruswami

Improved Maximally Recoverable LRCs using Skew Polynomials

An $(n,r,h,a,q)$-Local Reconstruction Code is a linear code over $\mathbb{F}_q$ of length $n$, whose codeword symbols are partitioned into $n/r$ local groups each of size $r$. Each local group satisfies `$a$' local parity checks to recover from `$a$' erasures in that local group and there are further $h$ global parity ... more >>>

TR21-036 | 14th March 2021
Siddharth Bhandari, Prahladh Harsha, Mrinal Kumar, Madhu Sudan

Ideal-theoretic Explanation of Capacity-achieving Decoding

In this work, we present an abstract framework for some algebraic error-correcting codes with the aim of capturing codes that are list-decodable to capacity, along with their decoding algorithm. In the polynomial ideal framework, a code is specified by some ideals in a polynomial ring, messages are polynomials and their ... more >>>

TR21-163 | 19th November 2021
Siddharth Bhandari, Prahladh Harsha, Mrinal Kumar, A. Shankar

Algorithmizing the Multiplicity Schwartz-Zippel Lemma

Revisions: 1

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 nu- merous applications in both math and computer ... more >>>

TR22-028 | 23rd February 2022
Dan Karliner, Roie Salama, Amnon Ta-Shma

The plane test is a local tester for Multiplicity Codes

Multiplicity codes are a generalization of RS and RM codes where for each evaluation point we output the evaluation of a low-degree polynomial and all of its directional derivatives up to order $s$. Multi-variate multiplicity codes are locally decodable with the natural local decoding algorithm that reads values on a ... more >>>

TR22-073 | 18th May 2022
Itay Kalev, Amnon Ta-Shma

Unbalanced Expanders from Multiplicity Codes

In 2007 Guruswami, Umans and Vadhan gave an explicit construction of a lossless condenser based on Parvaresh-Vardy codes. This lossless condenser is a basic building block in many constructions, and, in particular, is behind the state of the art extractor constructions.

We give an alternative construction that is based on ... more >>>

TR22-078 | 8th May 2022
Dan Karliner, Amnon Ta-Shma

Improved local testing for multiplicity codes

Multiplicity codes are a generalization of Reed-Muller codes which include derivatives as well as the values of low degree polynomials, evaluated in every point in $\mathbb{F}_p^m$.
Similarly to Reed-Muller codes, multiplicity codes have a local nature that allows for local correction and local testing.
Recently, the authors and ... more >>>

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