Posted by: atri | February 1, 2009

In Friday’s lecture, we saw the dual of Hamming codes, called Simplex code and the related Hamming code. We proved that Hamming codes satisfy $R=o(1)$ and $\delta=1/2$. We then looked at the following instantiations/special cases of the main motivating question for the worst-case noise model:

1. Can we achieve $R\ge \Omega(1)$ and $\delta\ge \Omega(1)$ simultaneously?
2. Can we have $R\ge \Omega(1)$ and $\delta>1/2$?
3. What is the optimal trade-off between $R$ and $\delta$?

We will see that the answer to (1) is yes, the answer to (2) is yes if $q>2$ and the answer to (3) is not know for small alphabets.

To see a concrete answer for (1), we saw the statement of the Gilbert bound and also some tight bounds on the volume of a Hamming ball.

For the relevant lecture notes from fall 07, see Lecture 7.