Minimal linear codes are in one-to-one correspondence with special types of blocking sets of projective spaces over a finite field, which are called strong or cutting blocking sets. In this paper we prove an upper bound on the minimal length of minimal codes of dimension $k$ over the $q$-element Galois field which is linear in both $q$ and $k$, hence improve the previous superlinear bounds. This result determines the minimal length up to a small constant factor. We also improve the lower and upper bounds on the size of so called higgledy-piggledy line sets in projective spaces and apply these results to present improved bounds on the size of covering codes and saturating sets in projective spaces as well. The contributions rely on geometric and probabilistic arguments.

翻译：最小线性代码在一对一的通信中,与在有限场域上特别类型的阻断投影空间的一组阻断装置有特殊类型,这些装置被称为强屏蔽装置或截断装置。在本文中,我们证明,在以美元和美元为线性的Galois $q $- element 字段上,最低维度代码的最小长度为 美元和 $ $ 美元,从而改进了先前的超线性界限。这个结果决定了最小的长度,直到一个小的恒定系数。我们还改进了投影空间上下限,并将这些结果用于显示在投影空间的覆盖码和饱和装置的大小。 贡献还取决于几何参数和概率参数。