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 of errors with rate for any desired constant .

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

We then saw how to run the -variate algorithm on a folded RS code with folding parameter , to list decode when the agreement fraction is more than . Picking large and , say and , then enables list decoding from agreement fraction . 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 and use higher degrees for the ‘s in the polynomial 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 , with restricting to be linear in the ‘s.

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

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