Prove that such a code does not exist

Question

I'm trying to solve the following exercise, but don't know what results to use.

Prove that it is not possible to find a linear code $\mathcal{C}[8, 5, 4]$ over $\mathbb{F}_2$ (without using the parity check matrix). What about the existence of a code $\mathcal{C}[8, 5, 3]$?

Answer 1

There is no linear $[2^r,2^r-r,3]$-code over $\Bbb F_2$ because of the sphere packing bound. Indeed, we would obtain $$ 2^r\ge \binom{2^r}{0}+\binom{2^r}{1}=2^r+1, $$ which is clearly false.

So there is no linear $[8,5,3]_2$-code.