关于 $\mathbb{F}^n_q$ 的循环覆盖子空间
On cyclically covering subspaces of $\mathbb{F}^n_q$
AI总结：
本文研究了 $\mathbb{F}^n_q$ 中循环覆盖子空间的性质，给出了 $ h_q(n) = 0 $ 的条件，并探讨了其在不同参数下的应用。
摘要：
对于素数幂 $ q $ 和正整数 $ n $，若子空间 $ U \subseteq \mathbb{F}_q^n $ 的所有循环移位的并集覆盖整个空间 $ \mathbb{F}_q^n $，则称该子空间为循环覆盖子空间。令 $ h_q(n) $ 表示此类子空间的最大可能余维数。本文聚焦于 $ h_q(n) = 0 $ 的情况。我们给出了 $ h_q(n) = 0 $ 成立的必要且充分条件。作为应用，我们证明当 $ q $ 是 $ \ell^t $ 的原根时，$ h_q(\ell^t) = 0 $。此外，我们证明若 $ n $ 为奇数且 $ h_q(n) = 0 $，则 $ h_q(2n) = 0 $。作为例子，我们展示 $ h_3(11) = h_3(16) = 1 $。此外，我们研究了 $\mathbb{F}_{q^m}^n$ 与 $\mathbb{F}_q^{mn}$ 的覆盖关系，并获得了若干 $ h_{q^m}(n) = 0 $ 的充分条件。具体地，我们推导出若 $ n = 3 $ 或 $ n = 2^d $（其中 $ d $ 为非负整数），则 $ h_4(n) = 0 $。
英文摘要：
For a prime power \( q \) and a positive integer \( n \), a subspace \( U \subseteq \mathbb{F}_q^n \) is called cyclically covering if the union of all its cyclic shifts covers the whole space \( \mathbb{F}_q^n \). Let \( h_q(n) \) denote the maximum possible codimension of such a subspace. This paper focuses on the case \( h_q(n) = 0 \). We provide necessary and sufficient conditions under which \( h_q(n) = 0 \) holds. As an application, we show that \( h_q(\ell^t) = 0 \) whenever \( q \) is a primitive root modulo \( \ell^t \). Moreover, we prove that if \( n \) is odd and \( h_q(n) = 0 \), then also \( h_q(2n) = 0 \). As an example, we show that \( h_3(11) =h_3(16) = 1 \). Furthermore, we investigate the relationship between the coverings of \(\mathbb{F}_{q^m}^n\) and \(\mathbb{F}_q^{mn}\), and obtain several sufficient conditions for \(h_{q^m}(n) = 0\). Specifically, we derive that if \(n = 3\) or \(n = 2^d\) (where \(d\) is a nonnegative integer), then \(h_4(n) = 0\).
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