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.