I encounter the following two exercises when learning coding theory, but I can't get a proof.
If there exists a $q$-ary code $(n,K,d)$, where $d=2l$ is an even number, prove that $q^n\geq K(q-1)\sum_{i=0}^{l-1}\binom{n}{i}(q-1)^i$
And
Suppose that $d$ is an even number, $2\leq d\leq n$. Prove that there exists a $(n,K,d)$ binary code iff there exists a $(n-1,K,d-1)$ binary code.