# Introduction to coding theory (CMU, Spr 2010)

## April 2, 2010

### Lecture 19 summary

Filed under: Lecture summary — Venkat Guruswami @ 7:54 pm

We pinned down the “list decoding capacity” for $q$-ary codes as $1-h_q(p)$, and saw that for large $q$ this implies codes of rate $R$ list-decodable up to a fraction $1-R-\epsilon$ of errors. We recalled the Johnson bound and how it implies a way to argue about list decoding properties knowing only the distance of the code. For Reed-Solomon codes this suggests that list-decoding from a fraction $1-\sqrt{R}$ of errors should be possible. This trade-off between rate and list decoding radius is best possible if one only appeals to the Johnson bound.

We then began the algorithmic segment of list decoding, setting up the stage for list decoding Reed-Solomon codes. In particular, we gave an algorithm for a toy problem where given a string which is the “mixture” of two codewords encoding polynomials $p_1$ and $p_2$, we recover both $p_1$ and $p_2$.