Download ALGORITHMIC RESULTS IN LIST DECODING (Foundations and by Venkatesan Guruswami PDF

By Venkatesan Guruswami

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

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1) by solving this linear system. The following simple lemma shows ˜ that we find. 1. If p(X) is a polynomial of degree at most k < n/6 such ˜ p(X)) ≡ 0, that p(αi ) = yi for at least n/3 values of i ∈ [n], then Q(X, ˜ or in other words Y − p(X) is a factor of Q(X, Y ). def Proof. Define the univariate polynomial R(X) = Q(X, p(X)). Let S = {i | p(αi ) = yi ; 1 i n}. For each i ∈ S, we have R(αi ) = ˜ i , yi ) = 0. Since the αi s are distinct, R(X) has at ˜ i , p(αi )) = Q(α Q(α least |S| n/3 roots.

10 in fact gives a list recovering algorithm for Reed–Solomon codes. ) Specifically, given input lists of size at most for each position of an RS code of block length n and dimension k + 1, √ the algorithm can find all codewords with agreement on more than kn positions. In other words, for any integer 1, an RS code of rate R and block length n√can be (p, , O(n2 2 ))-list-recovered in polynomial time when p < 1 − R . Note that the algorithm can do list recovery even in the noise-free (p = 0) case, only when R < 1/ .

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 . Then there exists a code C of block length n and rate r/d over γε2 alphabet Σd which is explicitly specified given C, H, and which is (1 − 1 − ε, L)-list-decodable in time O(T (n) + n). Furthermore, C is lineartime encodable if C is. 2. A toy list recovering problem 153 Proof. , A, B are both copies of the vertex set of H, and we connect u ∈ A with v ∈ B if (u, v) is an edge of H).

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