I have known how to calculate the Hamming distance between two message codes. But I don't know how to get the error control capability. In one case I have hamming distance of: $$ d = 8 $$
Errors detectable = $ d-1 = 7$
Errors correctable = $(\dfrac{d}{2}-1) = 3$
I don't know to finish it off from here.
Looking at another case which had $d=6$
Error detectable = 5
Error correctable = 2
....and then it goes further to say that:
because error control capability=5, 2-bits are used for: error correction and 3 bits for error detection, hence d=6 is DEC+TED.
How are these derived, precisely? How do the 3bits come about?
It looks like they are just allocating the check bits between correction and detection. A code with distance $d$ can correct errors up to $\lfloor \frac {d-1}2\rfloor = c$ bits. This agrees with your expression when $d$ is even but is one half greater when $d$ is odd. You can detect $d-1$ bit errors, so they are saying you have $c$ bits for correction and $d-1-c$ bits for detection.