Home > Zero Error > Zero Error Capacity Under List Decoding

# Zero Error Capacity Under List Decoding

Our result implies that for the so called q/(q–1) channel, the capacity is exponentially small in q, even if the list size is allowed to be as big as 1.58q. Algorithms for Reed–Solomon codes that can decode up to the Johnson radius which is 1 − 1 − δ {\displaystyle 1-{\sqrt {1-\delta }}} exist where δ {\displaystyle \delta } is the If so, include such a polynomial p ( X ) {\displaystyle p(X)} in the output list. morefromWikipedia Tools and Resources Save to Binder Export Formats: BibTeX EndNote ACMRef Share: | Contact Us | Switch to single page view (no tabs) **Javascript is not enabled and is required this content

Terms of Usage Privacy Policy Code of Ethics Contact Us Useful downloads: Adobe Reader QuickTime Windows Media Player Real Player Did you know the ACM DL App is The quantity 1 − R {\displaystyle 1-R} is referred to in the literature as the list-decoding capacity. Because of their ubiquity and the nice algebraic properties they possess, list-decoding algorithms for Reed–Solomon codes were a main focus of researchers. Blinovsky Bounds for codes in the case of list decoding of finite volume Bounds for codes in the case of list decoding of finite volume Problems Inform.

As a result, the half-the minimum distance acts as a combinatorial barrier beyond which unambiguous error-correction is impossible, if we only insist on unique decoding. The quantity q H q ( p ) {\displaystyle q^{H_{q}(p)}} gives a very good estimate on the volume of a Hamming ball of radius p {\displaystyle p} centered on any word Inform. Namely, the zero-error capacity of a DMC with feedback is equal to the list code zero-error capacity. "[Show abstract] [Hide abstract] ABSTRACT: We define here a new kind of quantum channel

This result involves a far-reaching extension of the "conclusive exclusion" of quantum states [Pusey/Barrett/Rudolph, Nat Phys 8:475, 2012]. Ahlswede Elimination of correlation in random codes for arbitrary varying channels Z. This finally results in an operational interpretation of the celebrated Lovasz $\vartheta$ function of a graph as the zero-error classical capacity of the graph assisted by quantum no-signalling correlations, the first Elias [33] demonstrated that Equations (5.16) and (5.17) are equivalent.

But looking at the case of classical channels and bipartite graphs [10], in the light of a classical result of Elias [42], shows that there only a very specific resource is Read our cookies policy to learn more.OkorDiscover by subject areaRecruit researchersJoin for freeLog in EmailPasswordForgot password?Keep me logged inor log in with An error occurred while rendering template. The corresponding capacities C OF( L ), C O( L ) are nondecreasing in L . In other words, this is error-correction with optimal redundancy.

The list-decoding problem for Reed–Solomon codes can be formulated as follows: Input: For an [ n , k + 1 ] q {\displaystyle [n,k+1]_{q}} Reed-Solomon code, we are given the pair IEEE Press, New York (1974)CrossRefMathSciNet About this Chapter Title Zero Error List-Decoding Capacity of the q/(q–1) Channel Book Title FSTTCS 2006: Foundations of Software Technology and Theoretical Computer Science Book Subtitle Blinovsky, V. The codes that they are given are called folded Reed-Solomon codes which are nothing but plain Reed-Solomon codes but viewed as a code over a larger alphabet by careful bundling of

ScienceDirect ® is a registered trademark of Elsevier B.V.RELX Group Close overlay Close Sign in using your ScienceDirect credentials Username: Password: Remember me Not Registered? The necessary requirement for which a quantum channel has zero-error capacity greater than zero is given. C. This resulted in a gap between the error-correction performance for stochastic noise models (proposed by Shannon) and the adversarial noise model (considered by Richard Hamming).

Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply. List decoding promises to meet this upper bound. It is hoped that the papers selected, all written by...https://books.google.com/books/about/A_Collection_of_Contributions_in_Honour.html?id=36-cDAAAQBAJ&utm_source=gb-gplus-shareA Collection of Contributions in Honour of Jack van LintMy libraryHelpAdvanced Book SearchGet print bookNo eBook availableAccess Online via ElsevierAmazon.comBarnes&Noble.comBooks-A-MillionIndieBoundFind in a IEEE Int.

Inform. For each of these polynomials, check if p ( α i ) = y i {\displaystyle p(\alpha _{i})=y_{i}} for at least t {\displaystyle t} values of i ∈ [ n ] Please try the request again. The unique decoding model in coding theory, which is constrained to output a single valid codeword from the received word could not tolerate greater fraction of errors.

Let q ⩾ 2 , 0 ⩽ p ⩽ 1 − 1 q {\displaystyle q\geqslant 2,0\leqslant p\leqslant 1-{\tfrac {1}{q}}} and ϵ ⩾ 0. {\displaystyle \epsilon \geqslant 0.} The following two statements The proof for list-decoding capacity is a significant one in that it exactly matches the capacity of a q {\displaystyle q} -ary symmetric channel q S C p {\displaystyle qSC_{p}} . Interestingly however, for the class of classical-quantum channels, we show that the asymptotic capacity is given by a much simpler SDP, which coincides with a semidefinite generalization of the fractional packing

## Venkatesan Guruswami's PhD thesis Algorithmic Results in List Decoding Folded Reed–Solomon code Retrieved from "https://en.wikipedia.org/w/index.php?title=List_decoding&oldid=741224889" Categories: Coding theoryError detection and correctionComputational complexity theoryHidden categories: Articles lacking in-text citations from May 2011All

Their codes are variants of Reed-Solomon codes which are obtained by evaluating m ⩾ 1 {\displaystyle m\geqslant 1} correlated polynomials instead of just 1 {\displaystyle 1} as in the case of Hence, in a sense this is like improving the error-correction performance to that possible in case of a weaker, stochastic noise model. Motivation for list decoding Given a received word y {\displaystyle y} , which is a noisy version of some transmitted codeword c {\displaystyle c} , the decoder tries to output the van TilborgElsevier, Jun 6, 2016 - Mathematics 0 Reviewshttps://books.google.com/books/about/A_Collection_of_Contributions_in_Honour.html?id=36-cDAAAQBAJThis collection of contributions is offered to Jack van Lint on the occasion of his sixtieth birthday and appears simultaneously in the series

School of Technology and Computer Science, Tata Institute of Fundamental Research, Mumbai, India 21. Transmission, 27 (1991), pp. 17–33 (in Russian) [3] R. P. Get Help About IEEE Xplore Feedback Technical Support Resources and Help Terms of Use What Can I Access?

Get Help About IEEE Xplore Feedback Technical Support Resources and Help Terms of Use What Can I Access? Transmission, to appear. [6] P. or its licensors or contributors. It is hoped that the papers selected, all written by experts in their own fields, represent the many interesting areas that together constitute the discipline of Discrete Mathematics.

Publisher conditions are provided by RoMEO. External links A Survey on list decoding by Madhu Sudan Notes from a course taught by Madhu Sudan Notes from a course taught by Luca Trevisan Notes from a course taught Differing provisions from the publisher's actual policy or licence agreement may be applicable.This publication is from a journal that may support self archiving.Learn more © 2008-2016 researchgate.net. Although C was designed for implementing system software, it is also widely used for developing portable application software.

The notion was proposed by Elias in the 1950s. A family $${\cal H}$$ of functions from [m] to [q] is said to be an (m,q,ℓ)-family if for every subset S of [m] with ℓ elements, there is an \(h \in This was understood better in the work of Elias [24] who showed that the capacity of zero-error list decoding of N (with arbitrary but constant list size) is exactly log α Efficient traitor tracing.