If there exists a q-tuple code (n, K, d) where $d = 2l$ is even, prove that
$q^n\ge K(q-1)\sum_{i=0}^{l-1}\binom{n}{i}(q-1)^i$
My thought was to take into account the number of vectors with distance $n$ from the code word, but that would be duplicative. Besides that, formally I don't know how this coefficient $(q-1)$ should be constructed
This question arises after the textbook proofs of Hamming Bound and Singleton Bound; perhaps the idea of a proof of Singleton Bound would be helpful for that question?