Posted by: atri | September 10, 2007

Lecture 6: General Hamming codes

In today’s lecture we saw two alternate characterizations of the distance of a linear code. We also defined the general Hamming code and verified that it was a perfect code.

Before the alarm went off, I was taking about the perfect binary linear codes. In fact the only possible family of such codes are:

• Hamming code
• $(n,1,n)_2$ code for odd $n$. (Note that the only codewords are $0^n$ and $1^n$.
• A $(23,12)_2$ code defined by Golay.

The result was proven in a sequence of work including those by van Lint and Tietavainen.