ALGORITHMIC RESULTS IN LIST DECODING (Foundations and by Venkatesan Guruswami

By Venkatesan Guruswami

Algorithmic ends up in record deciphering introduces and motivates the matter of checklist deciphering, and discusses the principal algorithmic result of the topic, culminating with the new effects on attaining "list interpreting capacity." the most technical concentration is on giving a whole presentation of the hot algebraic effects attaining record interpreting skill, whereas tips 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 machine technological know-how and knowledge idea. the writer concludes through posing a few fascinating 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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3. Suppose Q(X, Y ) is a nonzero polynomial with (1, k)weighted degree at most D satisfying Q(αi , yi ) = 0 for every i ∈ [n]. Let p(X) be a polynomial of degree at most k such that p(αi ) = yi for at least t > D values of i ∈ [n]. Then Q(X, p(X)) ≡ 0, or in other words Y − p(X) is a factor of Q(X, Y ). In view of the above lemma, if we could interpolate a nonzero polynomial Q(X, Y ) of (1, k)-weighted degree less than t satisfying 136 Decoding Reed–Solomon Codes Q(αi , yi ) = 0 for all i ∈ [n], then we can find all polynomials that have agreement at least t with the points amongst the factors of Q(X, Y ).

XΓd (j) . In other words, we “send” a copy of each symbol xi along all edges going out of the vertex i ∈ A, and the symbol yj is obtained by concatenating all symbols “received” by j ∈ B. The code G(C) is now obtained by taking all vectors G(c) for c ∈ C. 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. [33] Let ε, γ ∈ (0, 1) be arbitrary constants.

In turn this means that the alphabet size of RS codes must grow with the block length. The benefit of AG codes is that in general, algebraic curves over Fq can have many more rational points, in fact as many points as one needs. Thus one can define an infinite family of codes of increasing block lengths over a fixed alphabet Fq . The quality of the codes depends on the quality of the curve – as a curve contains more and more rational points, it must necessarily get more “twisted” and complicated, a phenomenon which is quantified by the genus of the curve.

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