What is the procedure into showing codewords are a perfect block code?

Question

Suppose that I had 2 given codewords i.e. 000 and 111.

I want to show that this is a perfect block code (or at least check).

Could anyone please outline the process of what I would do.

Thanks

Answer 1

It's perfect $1$-error correcting because the (closed) Hamming-balls of radius $1$ around the codewords partition the total message space:

The $1$-ball around $000$ is $\{000,001,010,100\}$ and the $1$-ball around $111$ is $\{111, 110, 101,011\}$, which are disjoint and cover $\{0,1\}^3$