  In recent years, many connections have been made between minimal codes, a
classical object in coding theory, and other remarkable structures in finite
geometry and combinatorics. One of the main problems related to minimal codes
is to give lower and upper bounds on the length $m(k,q)$ of the shortest
minimal codes of a given dimension $k$ over the finite field $\mathbb{F}_q$. It
has been recently proved that $m(k, q) \geq (q+1)(k-1)$. In this note, we prove
that $\liminf_{k \rightarrow \infty} \frac{m(k, q)}{k} \geq (q+ \varepsilon(q)
)$, where $\varepsilon$ is an increasing function such that $1.52
<\varepsilon(2)\leq \varepsilon(q) \leq \sqrt{2} + \frac{1}{2}$. Hence, the
previously known lower bound is not tight for large enough $k$. We then focus
on the binary case and prove some structural results on minimal codes of length
$3(k-1)$. As a byproduct, we are able to show that, if $k = 5 \pmod 8$ and for
other small values of $k$, the bound is not tight.
