Posted by: atri | February 1, 2009

Lecture 8: Hadamard Codes

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.

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