Introduction to coding theory (CMU, Spr 2010)

April 14, 2010

Lecture 22 summary

Filed under: Lecture summary — Venkat Guruswami @ 2:59 pm

We discussed how folded Reed-Solomon codes can be used to approach the optimal trade-off between rate and list decoding radius, specifically list decoding in polynomial time from a fraction 1-R-\epsilon of errors with rate R for any desired constant \epsilon > 0.

We presented an algorithm for list decoding folded Reed-Solomon codes (with folding parameter s) when the agreement fraction is more than \frac{1}{s+1} + \frac{s^2 R}{s+1}.  This was based on the extension of the Welch-Berlekamp algorithm to higher order interpolation (in s+1 variables). Unfortunately, this result falls well short of our desired target, and in particular is meaningless for R > 1/s.

We then saw how to run the (s+1)-variate algorithm on a folded RS code with folding parameter m > s, to list decode when the agreement fraction is more than \frac{1}{s+1} + \frac{s}{s+1} \frac{m}{m-s+1} R. Picking s large and m \gg s, say s \approx 1/\epsilon and m \approx 1/\epsilon^2, then enables list decoding from agreement fraction R+\epsilon. We will revisit this final statement briefly at the beginning of the next lecture, and also comment on the complexity of the algorithm, bound on list-size, and alphabet size of the codes.

Notes for this lecture may not be immediately available, but you can refer to the original paper Explicit codes achieving list decoding capacity: Error-correction with optimal redundancy or Chapter 6 of the survey Algorithmic results for list decoding.  Both of these are tailored to list decode even from the (in general) smaller agreement fraction \left(\frac{mR}{m-s+1}\right)^{s/(s+1)} and use higher degrees for the Z_i‘s in the polynomial Q(X,Z_1,\ldots,Z_s) as well as multiple zeroes at the interpolation points. In the lecture, however, we were content, for sake of simplicity and because it suffices to approach agreement fraction R + \epsilon, with restricting Q to be linear in the Z_i‘s.

A reminder that we will have NO lecture this Friday (April 16) due to Spring Carnival.


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