Download ALGORITHMIC RESULTS IN LIST DECODING (Foundations and by Venkatesan Guruswami PDF
By Venkatesan Guruswami
Algorithmic leads to record interpreting introduces and motivates the matter of checklist interpreting, 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 new algebraic effects attaining record deciphering potential, whereas tips or short descriptions are supplied for different works on record deciphering. Algorithmic ends up in record deciphering is meant for students and graduate scholars within the fields of theoretical computing device technology and knowledge idea. the writer concludes via 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)
Indeed, we can “unfold” the received word of the folded RS code and treat it as a received word of the original RS code and run the RS list decoding algorithm on it. The resulting list will certainly include all folded RS codewords within distance p of the received word, and it may include some extra codewords which we can, of course, easily prune. In fact, decoding the folded RS code is a strictly easier task. To see why, say we want to correct a fraction 1/4 of errors. , corrupts f (x4i ) for 0 i < n/4).
This suggests that we should make passing through each point multiple times an explicit requirement on the interpolated polynomial. Of course, this stronger property cannot be enforced for free and would require an increase in the degree of the interpolated polynomial. But luckily it turns out that one can pass through each point√twice with less than a two-fold increase in degree (a factor of roughly 3 suﬃces), and the trade-oﬀ between multiplicities guaranteed vs. degree increase is a favorable one.
This generalization is due to Guruswami and Sudan . 2. [Multiplicity of zeroes] A polynomial Q(X, Y ) is said to have a zero of multiplicity r 1 at a point (α, β) ∈ F2 if Q(X + α, Y + β) has no monomial of degree less than r with a nonzero coeﬃcient. 3 that takes multiplicities into account. 6. Let Q(X, Y ) be a nonzero polynomial of (1, k)-weighted degree at most D that has a zero of multiplicity r at (αi , yi ) 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/r values of i ∈ [n].