Introduction to coding theory (CMU, Spr 2010)

April 9, 2010

Lecture 21 summary

Filed under: Lecture summary — Venkat Guruswami @ 2:54 pm

Today we completed the description and analysis of the multiplicities based weighted polynomial reconstruction algorithm which immediately yielded an algorithm for list decoding Reed-Solomon codes up to the Johnson radius of $1-\sqrt{R}$ for rate $R$. We discussed the utility of weights in exploiting “soft” information available during decoding (eg. from decoding inner codes in a concatenation scheme, or from a demodulator which “rounds” analog signals to digital values). We saw simple consequences for list decoding binary concatenated codes, and in particular how to list-decode from a fraction $(1/2-\gamma)$ of errors with $\Omega(\gamma^6)$ rate and list-size $O(1/\gamma^3)$. While the rate is positive for every $\gamma > 0$, it is far from the optimal $\gamma^2$ bound. (We will soon see how this can be improved substantially by using codes with more powerful list-decoding properties than Reed-Solomon codes at the outer level.) Finally we defined (a version of) folded Reed-Solomon codes (we will give a list decoding algorithm for these next lecture).

We will have notes for this week’s lecture available soon, but the material covered this week has also been written about in several surveys on list decoding (some of which are listed on the course webpage). Here are a couple of pointers, which also discuss the details of list decoding folded RS codes which we will cover next week (though we will use a somewhat simpler presentation with weaker bounds in our lectures):

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