Posted by: atri | January 18, 2009

## Lecture 3: Distance of a code

In Friday’s lecture, we defined the minimum distance of a code and looked at its close connection to the error correction capability of the code. In particular, we saw that $\lfloor (d-1)/2\rfloor$ is a tight limit on the number of  errors that a code with distance $d$ can handle. (We also saw the limit for error detection and correcting erasures, where the limit turns out to be $d-1$.) We had seen special cases of these limits for the repetition and parity code earlier.

The result above implies that to figure out the maximum rate one can achieve for a given error correcting capability, one needs to find out the maximum rate achievable for a fixed distance. As a special case of this question we started to look at the question when the distance is $3$. We had already seen that the repetition code $C_{rep,3}$ has rate $1/3$. Towards the end of the class we talked about the Hamming code with rate $4/7$ that also has distance $3$.

The material covered in this lecture can be found in lecture 3 and lecture 4 notes from the fall 07 offering.

As $d$ is odd, $\lfloor (d-1)/2\rfloor$ is the same as $(d-1)/2$, so the expression in the notes is also OK.