By Venkatesan Guruswami

ISBN-10: 1601980043

ISBN-13: 9781601980045

Algorithmic ends up in checklist interpreting introduces and motivates the matter of record deciphering, and discusses the primary 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 new algebraic effects attaining checklist interpreting capability, whereas guidelines or short descriptions are supplied for different works on checklist interpreting. Algorithmic leads to checklist deciphering is meant for students and graduate scholars within the fields of theoretical laptop technology and knowledge idea. the writer concludes by way of posing a few fascinating open questions and indicates instructions for destiny paintings.

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**Additional resources for ALGORITHMIC RESULTS IN LIST DECODING (Foundations and Trends(R) in Theoretical Computer Science)**

**Example text**

3 List recovering from small but positive noise rate We now turn to the list recovering problem when the input list size is greater than 2, and more signiﬁcantly when we allow a small fraction of lists to be erroneous. 2. For every integer 1, there exist R > 0, γ > 0, and a ﬁnite alphabet Σ for which there is an explicit family of codes of rate R( ) over alphabet Σ that are encodable as well as (γ , , )-listrecoverable in linear time. 1, one gets the following result of Guruswami and Indyk [33] on linear-time list-decodable codes for correcting any desired constant fraction of errors.

Sn , we can reconstruct the list of all codewords c ∈ C such that G(c)j ∈ Sj , j = 1, . . , n, in linear time. For each i ∈ A, j ∈ B, we deﬁne L(i, j) to be the set of symbols in Σ that Sj “suggests” as potential symbols for ci . More formally, L(i, j) contains symbols ak , such that a1 , . . , ad ∈ Sj and Γk (j) = i. Deﬁne Ki = ∩(i,j)∈E L(i, j). We assume that all Ki s are non-empty, since otherwise no codeword compatible with the Si s exists. Deﬁne I to be the set of indices i ∈ A such that Ki has size 1.

1 Constructions over smaller alphabets Graph-based constructions have also been useful in obtaining good listdecodable codes over smaller alphabets compared with purely algebraic codes. 2 with the following components C, G: • C is a rate Ω(ε) code that is (1/2, Θ(1/ε), Θ(1/ε))-list recoverable. Such a code can be constructed via an appropriate concatenation involving a rate Θ(ε) outer RS code, and a good list-recoverable inner code found by brute-force search (see [31] for details). 4. Other graph-based list decoding results 159 have a neighborhood that spans more than half the nodes on the left.

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