By Venkatesan Guruswami

ISBN-10: 1601980043

ISBN-13: 9781601980045

Algorithmic ends up in checklist deciphering introduces and motivates the matter of record interpreting, and discusses the principal algorithmic result of the topic, culminating with the hot effects on reaching "list interpreting capacity." the most technical concentration is on giving an entire presentation of the hot algebraic effects attaining checklist interpreting means, whereas guidelines or short descriptions are supplied for different works on checklist interpreting. Algorithmic ends up in record interpreting is meant for students and graduate scholars within the fields of theoretical laptop technological know-how and knowledge conception. the writer concludes via posing a few attention-grabbing open questions and indicates instructions for destiny paintings.

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Extra info for ALGORITHMIC RESULTS IN LIST DECODING (Foundations and Trends(R) in Theoretical Computer Science)

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The subgraph of HH that corresponds to the codeword c. In other words, V (c) contains the nodes in A that are compatible with c. The key fact, easily proved by induction, is that if (i, t) ∈ V (c), and (i , t ) ∈ A is reachable from (i, t) in HH, then (i , t ) ∈ V (c). Hence V (c) will be the union of the vertices in A that belong to some subset of connected components of HH. The fault-tolerance property of G can be shown to imply that one of these connected components must contain at least n/10 vertices.

It is also useful in decoding concatenated codes, where the inner decoder can provide weights along with the choices it outputs, which can then be used by a soft-decision decoder for the outer code. As mentioned in [40], the multiplicity based interpolation lends itself naturally to a soft-decision version, since the multiplicity required at a point can encode the importance of that point. Given weights wi,γ for positions i ∈ [n] and field elements α ∈ F, we set the multiplicity of the point (αi , γ) to be proportional to wi,γ .

3. Let z ∈ (Fm )N and let y ∈ Fn be the unfolded version of z. Let Q(X, Y1 , Y2 ) be any nonzero polynomial over F of (1, k, k)weighted degree at D which has a zero of multiplicity r at (γ i , yi , yi+1 ) D . Then for i = 0, 1, . . , n − 1. Let t be an integer such that t > (m−1)r every polynomial f (X) ∈ F[X] of degree at most k whose encoding according to FRSF,γ,m,k agrees with z on at least t locations satisfies Q(X, f (X), f (γX)) ≡ 0. 3 motivate the following approach to list decoding the folded RS code FRSF,γ,m,k .

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ALGORITHMIC RESULTS IN LIST DECODING (Foundations and Trends(R) in Theoretical Computer Science) by Venkatesan Guruswami

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