By Venkatesan Guruswami
Algorithmic ends up in checklist deciphering introduces and motivates the matter of checklist deciphering, and discusses the valuable algorithmic result of the topic, culminating with the hot effects on attaining "list deciphering capacity." the most technical concentration is on giving a whole presentation of the new algebraic effects attaining checklist interpreting means, whereas guidelines or short descriptions are supplied for different works on record deciphering. Algorithmic leads to checklist interpreting is meant for students and graduate scholars within the fields of theoretical laptop technological know-how and data thought. the writer concludes through posing a few attention-grabbing open questions and indicates instructions for destiny paintings.
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Additional info for ALGORITHMIC RESULTS IN LIST DECODING (Foundations and Trends(R) in Theoretical Computer Science)
The reduction is described formally below. Note that the error fraction is reduced from (1 − 1/ ) (which is close to 1 for large ) to γ for an arbitrarily small constant γ > 0 (at the expense of a larger alphabet size and lower rate). 1.  Let ε, γ ∈ (0, 1) be arbitrary constants. Let C be a code of block length n and rate r over alphabet Σ. Also suppose that C is (γ, , L)-list-recoverable in time T (n). Further, assume that there exists a d-regular Ramanujan expander H on n vertices for d 4 .
Let γ be a generator of the multiplicative group F∗ , and let the evaluation points be ordered as 1, γ, γ 2 , . . , γ n−1 . Using all nonzero ﬁeld elements as evaluation points is one of the most commonly used instantiations of Reed–Solomon codes. Let m 1 be an integer parameter called the folding parameter. For ease of presentation, we will assume that m divides n = q − 1. 1. (Folded Reed–Solomon code) The m-folded version of the RS code C, denoted FRSF,γ,m,k , is a code of block length N = n/m over Fm .
In this chapter, we will describe this combined code and algorithm. We note that this presentation deviates signiﬁcantly from the historical development in the original papers [37,62], in that we are using the beneﬁt of hindsight to give a self-contained, and hopefully simpler, presentation. The last section of this chapter contains more comprehensive bibliographic notes on the original development of this material. 1 Description of folded codes Consider a Reed–Solomon code C = RSF,F∗ [n, k] consisting of evaluations of degree k polynomials over F at the set F∗ of nonzero elements of F.