关于极小码长度的下界
On the lower bound for the length of minimal codes
摘要：
近年来，在编码理论中的经典对象极小码与有限几何和组合学中的其他显著结构之间已经建立了许多联系。与极小码有关的主要问题之一是给出有限域$\mathbb{F}_q$上给定维数$k$的最短极小码的长度$m(k,q)$的上下界.最近已证明$m(k, q) \geq (q+1)(k-1)$。本文证明了$\liminf_{k \rightarrow \infty} \frac{m(k, q)}{k} \geq (q+ \varepsilon(q) )$，其中$\varepsilon$是一个增函数，使得$1.52 <\varepsilon(2)\leq \varepsilon(q) \leq \sqrt{2} + \frac{1}{2}$ .因此，对于足够大的$k$，先前已知的下限并不严格。然后，我们专注于二进制的情况下，并证明了一些结构的最小码长度$3(k-1)$的结果。作为一个副产品，我们能够表明，如果$k = 5 \pmod 8$和其他小值的$k$，边界是不紧。
英文摘要：
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.
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