A Gilbert-type lower bound for a linear code

Given integers \(0<d\le n\) and a prime power \(q\), prove that there exists a linear code \[ C\subseteq\mathbb F_q^n \] of minimum distance at least \(d\), containing at least \[ \frac{q^n}{\displaystyle\sum_{i=0}^{d-1}\binom ni(q-1)^i} \] codewords.