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.

翻译：近年来，极小码（编码理论中的经典研究对象）与有限几何及组合数学中的其他显著结构之间建立了诸多联系。与极小码相关的主要问题之一是：给定域 $\mathbb{F}_q$ 上维度为 $k$ 的最短极小码长度 $m(k,q)$ 的下界与上界。近期已证明 $m(k, q) \geq (q+1)(k-1)$。本研究证明 $\liminf_{k \rightarrow \infty} \frac{m(k, q)}{k} \geq (q+ \varepsilon(q) )$，其中 $\varepsilon$ 为单调递增函数且满足 $1.52 <\varepsilon(2)\leq \varepsilon(q) \leq \sqrt{2} + \frac{1}{2}$。因此，当 $k$ 足够大时，先前已知的下界并非紧界。随后我们聚焦于二元情形，证明了长度为 $3(k-1)$ 的极小码的结构性质。作为副产品，我们发现当 $k \equiv 5 \pmod 8$ 及其他某些小 $k$ 值时，该下界仍非紧界。