By Venkatesan Guruswami
Algorithmic leads to checklist interpreting introduces and motivates the matter of record deciphering, and discusses the relevant algorithmic result of the topic, culminating with the hot effects on reaching "list deciphering capacity." the most technical concentration is on giving a whole presentation of the new algebraic effects reaching 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 desktop technological know-how and knowledge concept. the writer concludes by means of posing a few fascinating open questions and indicates instructions for destiny paintings.
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Extra resources for ALGORITHMIC RESULTS IN LIST DECODING (Foundations and Trends(R) in Theoretical Computer Science)
The following records the performance of this algorithm. Again, the size of the output list never exceeds the degree of Q(X, Y ) in Y , which is at most D/k . 8. 1 The size of the list output by the algorithm never exceeds nr(r + 1)/k. 9. A Reed–Solomon code of rate R can be list decoded in polynomial √ time up to a fraction 1 − (1 + ε)R of errors using lists −1 of size O(ε / R). By letting the multiplicity r grow with as long as √ n, we can decode 2 the agreement parameter satisﬁes t > kn. Indeed, if t > kn, picking r=1+ kn − t , t2 − kn and D = rt − 1, both the conditions t > Dr and D+2 > n r+1 are 2 2 satisﬁed, and thus the decoding algorithm successfully ﬁnds all polynomials with agreement at least t.
This is the main result of this chapter. Note √ that for every rate R, 0 < R < 1, the decoding radius 1 − R exceeds the best decoding radius (1 − R)/2 that we can hope for with unique decoding. 1 Let x = knr(r + 1). The condition t > D/r with D = [x] is implied by t > [x/r], since x/r [x]/r < [x/r] + 1. 4. Extensions: List recovering and soft decoding 145 Remark 5. [Role of multiplicities] Using multiplicities in the interpolation led to the √ improvement of the decoding radius to match the Johnson radius 1 − R and gave an improvement over unique decoding for every rate R.
5. The above algorithm solves the polynomial reconstruction problem in polynomial time if the agreement parameter t satisﬁes √ t > 2kn . The size of the list output by the algorithm never exceeds 2n/k. The above implies that√a Reed–Solomon code of rate R can be list decoded to a fraction 1 − 2R of errors using lists of size O( 1/R). In particular, for low-rate codes, we can eﬃciently correct close to a fraction 100% of errors! This qualitative feature is extremely useful in many areas of complexity theory, such as constructions of hardcore predicates from one-way functions, randomness extractors and pseudorandom generators, hardness ampliﬁcation, transformations from worst-case to average-case hardness, etc.