Posted by: atri | February 5, 2009

Lecture 10: Hamming Ball Volume and Singleton Bound

In today’s lecture we saw the following approximation for the volume of a $q$-ary Hamming ball of radius $pn, 0\le p\le 1-1/q$:

$q^{H_q(p)n-o(n)}\le Vol(q,pn)\le q^{H_q(p)n}$.

For the proof, see fall 07 notes on lecture 9 (Section 1.2).

We then proved the Singleton bound– for the proof see fall 07 notes on lecture 11 (Sec 3). We then saw that for large enough alphabet, the Gilbert-Varshamov bound approaches the Singleton bound. Next lecture, we will study Reed-Solomon codes which match the Singleton bound.

A quick poll for you guys: we are at a position where I can talk about some application (in theoretical computer science) of the material we have covered (or are going to cover very soon). Do you guys want to hear about those now or would rather not break the flow of studying better upper bound on the rate vs. distance question. Feel free to use the comments section to “vote.”