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.

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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Responses

  1. 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).

  2. Swapnoneel,

    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.


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