A few detection and correction questions

Question

I need a bit of help with error detection and correction.

We have:

 C = {01010101, 10101010, 00000000, 11111111}

I need to:

1) compute d(C)  = min ({Hd(v,w) / v, w belongs to C});
2) find an error that can’t be detected where Hw(v) <= (d(c)-1)/3;
3) find an error that can be detected but not corrected;

PS. Is not a homework, is just one basics subjects from an old exam.

It would help me a lot a few solving hints. What I actually need, is to understand how to do them.

Answer 1

1. You see that $C$ is a vector space, so $d(C)$ is just the minimum Hamming distance from zero of a non-zero vector in $C$, hence $4$.
2. Any error that takes an element of $C$ to another element of $C$, for instance when $01010101$ is sent, and $00000000$ is received.
3. Suppose you receive $v = 11110000$. This has distance $4$ from all the elements of $C$, so it could come via $4$ bit-errors from any of them. You see there are errors because $v \notin C$.