Posted by: atri | April 19, 2009

## Lecture 36: List Decoding of RS codes

In Friday’s lecture, we started with the two natural questions in list decoding:

1. Can we achieve list decoding capacity with explicit codes and efficient decoding algorithms? In particular, one we achieve $p\ge 1-R-\epsilon$ for large enough alphabet size?
2. As a less aggressive goal, can we achive the Johnson bound, i.e. $p\ge 1-\sqrt{R}$, with efficient list decoding algorithms.

In this course, we will see positive answers to both questions with codes that are either RS codes or are generalizations of RS codes. Towards this end, we started with list decoding algorithms for RS codes. We started with the generalization of the Berlekamp-Welch algorithm by Sudan, which leads to an efficient list decoding algorithm that can correct $1-\sqrt{2R}$ fraction of errors. We did not specify one crucial aspect of the algorithm, which we will do in Monday’s lecture. We looked at an example that gave the intuition behind why Sudan’s algorithm. You can find the example in this survey.

Due to limited time, we skipped an easier version of Sudan’s list decoding algorithm that can correct up to $1-2\sqrt{R}$ fraction of errors. For more details on this, see the scribed notes of Lecture 37 from Fall 07. (The latter notes are not polished yet: hopefully they’ll be in a better shape by the end of the week.)