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Kopparty, Swastik

doi:10.4086/toc.2015.v011a005

Cited by 53 publications

(6 citation statements)

References 30 publications

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Section: Similarities With Prgs Of Shaltiel and Umans [Su05 Uma03]mentioning

confidence: 79%

“…We also note that the set up of induction we have in the proof of Lemma 3.2 is very similar to the set up used by Kopparty [Kop15] in the context of list decoding Multiplicity codes. More precisely, our induction is similar to what is used in constructing a power series expansion of a non-degenerate solution of the univariate Cauchy-Kovalevski differential equations, which are used in [Kop15]. The key difference is that although we work with a multivariate setting (and hence deal with a partial differential equation of high order and high degree), the iterative proof of Lemma 3.2 resembles the list decoding algorithm for univariate multiplicity codes in [Kop15] (which deals with an ordinary differential equation of high order and high degree).…”

Section: Similarities With Prgs Of Shaltiel and Umans [Su05 Uma03]mentioning

confidence: 99%

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Derandomization from Algebraic Hardness: Treading the Borders

Guo

Kumar

Saptharishi

et al. 2019

2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)

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A hitting-set generator (HSG) is a polynomial map Gen : F k → F n such that for all n-variate polynomials C of small enough circuit size and degree, if C is nonzero, then C • Gen is nonzero. In this paper, we give a new construction of such an HSG assuming that we have an explicit polynomial of sufficient hardness. Formally, we prove the following result over any field F of characteristic zero:Suppose P(z 1 , . . . , z k ) is an explicit k-variate degree d polynomial that is not computable by circuits of size s. Then, there is an explicit hitting-set generator Gen P :This is the first HSG in the algebraic setting that yields a complete derandomization of polynomial identity testing (PIT) for general circuits from a suitable algebraic hardness assumption. Unlike the prior constructions of such maps [NW94, KI04, AGS19, KST19], our construction is purely algebraic and does not rely on the notion of combinatorial designs.As a direct consequence, we show that even saving a single point from the "trivial" explicit, exponential sized hitting sets for constant-variate polynomials of low individual-degree which are computable by small circuits, implies a deterministic polynomial time algorithm for PIT. More precisely, we show the following:Let k be a large enough constant. Suppose for every s large enough, there is an explicit hitting set of size at most ((s + 1) k − 1) for the class of k-variate polynomials of individual degree s that are computable by size s circuits. Then there is an explicit hitting set of size s O(k 2 ) for the class of s-variate polynomials, of degree s, that are computable by size s circuits.

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Section: Similarities With Prgs Of Shaltiel and Umans [Su05 Uma03]mentioning

confidence: 79%

Section: Similarities With Prgs Of Shaltiel and Umans [Su05 Uma03]mentioning

confidence: 99%

Derandomization from Algebraic Hardness: Treading the Borders

Guo

Kumar

Saptharishi

et al. 2019

2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)

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“…For example uniformly random codes achieve capacity, as do uniformly random linear codes [21,40,20]. Folded Reed-Solomon codes were the first explicit codes to provably achieve list-decoding capacity [22], followed by several others a few years later [23,34,26,45,15]. For the rest of this paper however, we will exclusively work in the model where the errors are stochastic.…”

Section: List Decodingmentioning

confidence: 99%

A criterion for decoding on the binary symmetric channel

Rao,

Sprumont

2025

AMC

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We present an approach to showing that a linear code is resilient to random errors. We use this approach to obtain decoding results for both transitive and doubly transitive codes. We give three kinds of results about linear codes in general, and transitive linear codes in particular.1. We give a tight bound on the weight distribution of every transitive linear code2. We give a criterion that certifies that a linear code C can be decoded on the binary symmetric channel. Let Ks(x) denote the Krawtchouk polynomial of degree s, and let C ⊥ denote the dual code of C. We show that bounds on E c∈C ⊥ [K ϵN (wt(c)) 2 ] imply that C recovers from errors on the binary symmetric channel with parameter ϵ. Weaker bounds can be used to obtain list-decoding results using similar methods. One consequence of our criterion is that whenever the weight distribution of C ⊥ is sufficiently close to the binomial distribution in some interval around N 2 , C is resilient to ϵ-errors. 3. We combine known estimates for the Krawtchouk polynomials with our weight bound for transitive codes, and with known weight bounds for doubly transitive codes, to obtain list-decoding results for both these families of codes. In some regimes, our bounds for doubly transitive codes achieve the information-theoretic optimal trade-off between rate and list size.

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“…There are several known list-decodable codes with efficient decoding algorithms, with varying parameters. This includes codes which use algebraic structure, such as Reed--Solomon codes [29,16], folded Reed--Solomon codes [28,14], multiplicity codes [24], algebraic-geometric codes [16,17], and constructions using a more combinatorial approach, such as those in [11,13,12]. Most of these constructions get parameters better than our constructions, and some of them get very close to optimal rate and alphabet size.…”

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confidence: 92%

List-Decoding with Double Samplers

Dinur¹,

Harsha²,

Kaufman³

et al. 2021

SIAM J. Comput.

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We strengthen the notion of double samplers, first introduced by Dinur and Kaufman [``High dimensional expanders imply agreement expanders,"" in Proc. 58th IEEE Symp. on Foundations of Comp. Science, IEEE, 2017, pp. 974-985], which are samplers with additional combinatorial properties, and whose existence we prove using high-dimensional expanders. The ABNNR code construction [N. Alon et al., IEEE Trans. Inform. Theory, 38 (1992), pp. 509-516] achieves large distance by starting with a base code C with moderate distance, and then amplifying the distance using a sampler. We show that if the sampler is part of a larger double sampler, then the construction has an efficient list-decoding algorithm. Our algorithm works even if the ABNNR construction is not applied to a base code C but rather to any string. In this case the resulting code is approximate-list-decodable, i.e., the output list contains an approximation to the original input. Our list-decoding algorithm works as follows: It uses a local voting scheme from which it constructs a unique games constraint graph. The constraint graph is an expander, so we can solve unique games efficiently. These solutions are the output of the list-decoder. This is a novel use of a unique games algorithm as a subroutine in a decoding procedure, as opposed to the more common situation in which unique games are used for demonstrating hardness results. Double samplers and high-dimensional expanders are akin to pseudorandom objects in their utility, but they greatly exceed random objects in their combinatorial properties. We believe that these objects hold significant potential for coding theoretic constructions and view this work as demonstrating the power of double samplers in this context.

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Cited by 53 publications

References 30 publications

Derandomization from Algebraic Hardness: Treading the Borders

Derandomization from Algebraic Hardness: Treading the Borders

A criterion for decoding on the binary symmetric channel

List-Decoding with Double Samplers

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