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By Venkatesan Guruswami

Algorithmic leads to record deciphering introduces and motivates the matter of record deciphering, and discusses the important algorithmic result of the topic, culminating with the hot effects on attaining "list interpreting capacity." the most technical concentration is on giving a whole presentation of the hot algebraic effects attaining record deciphering ability, whereas guidelines or short descriptions are supplied for different works on record interpreting. Algorithmic ends up in record deciphering is meant for students and graduate scholars within the fields of theoretical laptop technology and knowledge thought. the writer concludes by way of posing a few attention-grabbing open questions and indicates instructions for destiny paintings.

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Z × {1, 2, . . , }, E) ˜ whose vertex set corgraph” H responds to the possible choices for the correct symbol at each node of Y, Z. The edge set is defined based on which symbol of L2 (i, j) matches the r-th symbol of Ki for r = 1, 2, . . , (as per some fixed ordering of the elements in each Ki and L2 (i, j)). The problem of finding a consis˜ tent codeword can then be cast as finding a very dense subgraph of H that internally closely resembles a copy of the expander G2 . Using the fact that this subgraph resembles G2 and that G2 is a high quality 158 Graph-Based List-Decodable Codes Ramanujan expander, Guruswami and Indyk [33] demonstrate how spectral techniques can be used to find such subgraphs and thus all the solution codewords in O(n log n) time.

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 . The encoding of a message f (X), a polynomial over F of degree at most k, has its j-th symbol, for 0 j < n/m, the m-tuple (f (γ jm ), f (γ jm+1 ), .

1. Description of folded codes 163 of high rate. Together, this gives explicit codes with polynomial time decoding algorithms that achieve list decoding capacity. 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.

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