AI 中文总结
本文研究了循环覆盖子空间的余维数，通过常循环码和不可约循环码的方法，推导了循环覆盖子空间的下界并给出了若干族n使得余维数为零。
AI 中文摘要
一个$\mathbb{F}_q^n$的子空间被称为循环覆盖的，如果$\sigma^i(U)$的并集可以覆盖整个空间$\mathbb{F}_q^n$，其中$\sigma$是循环移位，$0 \leqslant i \leqslant n-1$。令$h_q(n)$为$\mathbb{F}_q^n$中最大可能的循环覆盖子空间的余维数。我们证明对于每一个素数$p$，使得2是模$p$的原根，都有$h_2(2p) = 2$。通过常循环码，我们证明当$h_q(n) = 0$且$\gcd(n,q-1) = 1$时，有$h_q((q-1)n) = 0$。我们还通过支撑重量分布的概念推导了$h_q(n)$的一个下界，这在编码理论中很重要。最后，利用不可约循环码，我们给出了若干族$n$使得$h_q(n) = 0$。
英文摘要
A subspace of $\mathbb{F}_q^n$ is called cyclically covering if the union of $σ^i(U)$ can cover the whole space $\mathbb{F}_q^n$, where $σ$ is the cyclic shift, $0 \leqslant i \leqslant n-1$. Let $h_q(n)$ be the largest possible co-dimension of a cyclically covering subspace of $\mathbb{F}_q^n$. We show that $h_2(2p) = 2$ for every prime $p$ such that $2$ is a primitive root modulo $p$. By constacyclic codes, we show that $h_q((q-1)n) = 0$ when $h_q(n) = 0$ and $\gcd(n,q-1) = 1$. We also derive a lower bound on $h_q(n)$ by the concept of support weight distribution, which is important in coding theory. Finally, using irreducible cyclic codes, we present several families of $n$ such that $h_q(n) = 0$.
Comments20 pages
