Posted by: atri | April 9, 2009

## Lecture 32: Derandomizing GMD Decoder

In Wednesday’s lecture we first finished the proof to show that the randomized version of the GMD we considered in Monday’s lecture in expectation can correct up to half the design distance of certain concatenated code. We also derandomized the algorithm and on the way, we also saw another randomized version of the GMD decoding algorithm. The randomized GMD decoding algorithms are can be found in the scribed notes for Lecture 28 while the deterministic GMD decoder can be found in the scribed notes for  Lecture 29 from Fall 07. (The latter notes will be polished by this weekend.)

On Friday, we will construct an explicit linear concatenated code, along with efficient decoding algorithms that achieves the capacity on the $BSC_p$. As a warmup, I encourage you to think about this problem: given a linear code of rate $1-H(p)-\epsilon$ and block length $n$, show that one can verify whether the MLD on the code achieves exponentially small decoding error probability in time $2^{O(n)}$.