# Introduction to coding theory (CMU, Spr 2010)

## April 21, 2010

### Lecture 23 summary

Filed under: Lecture summary — Venkat Guruswami @ 3:29 pm

We completed the discussion of the rate vs. list decoding radius trade-off achieved by folded Reed-Solomon codes and multivariate interpolation based decoding, and discussed its complexity and list-size bounds, as well as alphabet size. We highlighted the powerful list recovery property offered by folded RS codes, where having up to $\ell$ possible choices for each codeword position does not affect the ability to correct with agreement $R + \epsilon$ (where $R$ is the rate), and we can “absorb” the effect of $\ell$ into a somewhat larger alphabet size and decoding complexity. This feature is invaluable in using folded RS codes as outer codes in concatenation schemes, as we saw in two results:

1. Binary codes which are list-decodable up to the Zyablov radius (earlier we saw to unique decode up to half the Zyablov radius using GMD decoding)
2. Construction of codes of rate $R$ over an alphabet of size $\exp((1/\epsilon)^{O(1)})$ that are list-decodable up to a fraction $1-R-\epsilon$ of errors. The alphabet size is not far from the optimal bound of $\exp(1/\epsilon)$, and nicely combines ideas from the algebraic coding and expander decoding parts of the course.

We then wrapped up our discussion of list decoding by mentioning some of the big questions that still remain open, especially in constructing binary codes with near-optimal (or even better than currently known) trade-offs.

We discussed the framework of message-passing algorithms for LDPC codes, which will be the subject of the next lecture or two. We will mostly follow the description in this survey, but will not get too deep into the material.