On cyclically covering subspaces of $\mathbb{F}^n_q$

Yangcheng Li; Pingzhi Yuan; Shuang Li; Yuanpeng Zeng

On cyclically covering subspaces of $\mathbb{F}^n_q$

Yangcheng Li, Pingzhi Yuan, Shuang Li, Yuanpeng Zeng

TL;DR

The paper studies cyclically covering subspaces of $\mathbb{F}_q^n$ by characterizing when $h_q(n)=0$ using an isomorphism-based framework that links $\mathbb{F}_q^n$, $\mathbb{F}_{q^n}$, and $\mathbb{F}_q[x]/(x^n-1)$. It derives equivalent conditions via invariant subspaces, trace-orthogonal constructions, and direct-sum decompositions, and applies these to obtain zero-codimension results such as $h_q(\ell^t)=0$ when $q$ is a primitive root modulo $\ell^t$, as well as a doubling property $h_q(2n)=0$ for odd $n$ when $h_q(n)=0$. Additionally, the work connects coverings in $\mathbb{F}_{q^m}^n$ to those in $\mathbb{F}_q^{mn}$, yielding transfer criteria and bounds, and provides concrete examples (e.g., $h_3(11)=h_3(16)=1$) along with sufficient conditions for $h_{q^m}(n)=0$ in various arithmetic settings, including $h_4(n)=0$ for $n=3$ or $n=2^d$. Overall, the results broaden the toolkit for understanding cyclic coverings and give practical criteria for zero codimension across different finite-field regimes.

Abstract

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 $h_4(n) = 0$.

On cyclically covering subspaces of $\mathbb{F}^n_q$

TL;DR

The paper studies cyclically covering subspaces of

by characterizing when

using an isomorphism-based framework that links

, and

. It derives equivalent conditions via invariant subspaces, trace-orthogonal constructions, and direct-sum decompositions, and applies these to obtain zero-codimension results such as

when

is a primitive root modulo

, as well as a doubling property

for odd

when

. Additionally, the work connects coverings in

to those in

, yielding transfer criteria and bounds, and provides concrete examples (e.g.,

) along with sufficient conditions for

in various arithmetic settings, including

for

. Overall, the results broaden the toolkit for understanding cyclic coverings and give practical criteria for zero codimension across different finite-field regimes.

Abstract

For a prime power

and a positive integer

, a subspace

is called cyclically covering if the union of all its cyclic shifts covers the whole space

. 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

is a primitive root modulo

. Moreover, we prove that if

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

and

, and obtain several sufficient conditions for $h_{q^m}(n) = 0$. Specifically, we derive that if

(where

is a nonnegative integer), then $h_4(n) = 0$.

Paper Structure (6 sections, 19 theorems, 78 equations, 1 table)

This paper contains 6 sections, 19 theorems, 78 equations, 1 table.

Introduction
Preliminaries
The equivalent conditions for $h_q(n)=0$
Coverings over $\mathbb{F}_{q^m}^n$ and $\mathbb{F}_q^{mn}$
Conclusions and further work
Appendix

Key Result

Lemma 1.1

Cameron-Ellis-Raynaud Let $q$ be a power of prime $p$, and $n,m,d,k \in \mathbb{N}$, then the following hold. $(\mathrm{i})$$h_2(n) \geq 2$, $n > 3$. $(\mathrm{ii})$$h_q(nm) \geq \max\{h_q(n), h_q(m)\}.$$(\mathrm{iii})$$h_q(n) \leq \lfloor\log_q(n)\rfloor$. $(\mathrm{iv})$$h_q(q^d - 1) = d - 1 = \lf

Theorems & Definitions (39)

Lemma 1.1
Lemma 1.2
Lemma 1.3
Lemma 1.4
Theorem 3.1
proof
Remark 3.1
Theorem 3.2
proof
Theorem 3.3
...and 29 more

On cyclically covering subspaces of $\mathbb{F}^n_q$

TL;DR

Abstract

On cyclically covering subspaces of $\mathbb{F}^n_q$

Authors

TL;DR

Abstract

Table of Contents

Key Result

Theorems & Definitions (39)