Posted by: atri | January 28, 2009

## Lecture 7: More on linear codes

In today’s lecture we started by looking at the solution to the “puzzle” I posed at the end of the last lecture: define the $i$th column of the parity check matrix to be the binary representation of $i$. This gives rise to the general Hamming code.

We then proved two results about the distance of any linear code. That is, the distance of a linear code is the same as the minimum Hamming weight of any non-zero codeword as well as the minimum number of linearly dependent column in the parity check matrix of the code. We used the later characterization to show that the general Hamming code has distance $3$ (and hence is a perfect code).

Recall that our main goal is to find the best possible relationship between the rate $R$ and the fraction of errors (which for the Hamming noise model is the same as the relative distance $\delta$ of the code). Hamming code gives a family that achieves $R= 1-o(1)$ and $\delta=o(1)$.
For the relevant Fall 07 lecture notes see lecture 6.

In the next lecture, we will see that the dual of the Hamming code has $\delta=\Omega(1)$.