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 is a tight limit on the number of errors that a code with distance can handle. (We also saw the limit for error detection and correcting erasures, where the limit turns out to be .) 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 . We had already seen that the repetition code has rate . Towards the end of the class we talked about the Hamming code with rate that also has distance .

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

Again, feel free to use the comments section for any question that you might have about this lecture and/or the course.

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There’s a minor mistake in lecture 3 (notes). When C has a distance d (d is odd) it can correct \lfloor (d-1)/2\rfloor errors. The “floor” is missing in the lecture notes (in proposition 2.3).

By:

Swapnoneel Royon January 18, 2009at 7:53 pm

Swapnoneel,

As is odd, is the same as , so the expression in the notes is also OK.

By:

atrion January 18, 2009at 10:08 pm