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On Trivial Cyclically Covering Subspaces of $\mathbb{F}_q^n$ in Non-Coprime Characteristic

Shuang Li, Pingzhi Yuan

TL;DR

The paper addresses the question of when h_q(n)=0 for non-coprime n, extending Huang’s complete coprime solution. It develops a modular group-algebra framework, expressing \mathbb{F}_q G as a deformation \mathbb{F}_q H[u]/\langle u^{p^k} \rangle and decomposing it into components A_t = \mathbb{F}_{q^{d_t}}[u]/\langle u^{p^k} \rangle, with a residue trace map connecting to Huang’s trace criterion. The main result shows that h_q(n)=0 holds if and only if h_q(m)=0 for n = p^k m with gcd(m, p)=1, effectively reducing the non-coprime case to the coprime classification. This provides a unified criterion for trivial cyclically covering subspaces across all n and highlights the role of the residue trace in translating modular structure to the field-trace conditions of Huang. The findings offer a complete classification tool and pave the way for further exploration of covering properties via modular representations and trace methods.

Abstract

A subspace $U$ of $\mathbb{F}_q^n$ is called \textit{cyclically covering} if the whole space $\mathbb{F}_q^n$ is the union of the cyclic shifts of $U$. The case $\mathbb{F}_q^n$ itself is the only covering subspace, is of particular interest. Recently, Huang solved this problem completely under the condition $\gcd(n, q)=1$ using primitive idempotents and trace functions, and explicitly posed the non-coprime case as an open question. This paper provides a complete answer to Huang's question. We prove that if $n = p^k m$ where $p = \operatorname{char}(\mathbb{F}_q)$ and $\gcd(m, p)=1$, then $h_q(p^k m) = 0$ if and only if $h_q(m) = 0$. This result fully reduces the non-coprime case to the coprime case settled by Huang. Our proof employs the structure theory of cyclic group algebras in modular characteristic.

On Trivial Cyclically Covering Subspaces of $\mathbb{F}_q^n$ in Non-Coprime Characteristic

TL;DR

The paper addresses the question of when h_q(n)=0 for non-coprime n, extending Huang’s complete coprime solution. It develops a modular group-algebra framework, expressing \mathbb{F}_q G as a deformation \mathbb{F}_q H[u]/\langle u^{p^k} \rangle and decomposing it into components A_t = \mathbb{F}_{q^{d_t}}[u]/\langle u^{p^k} \rangle, with a residue trace map connecting to Huang’s trace criterion. The main result shows that h_q(n)=0 holds if and only if h_q(m)=0 for n = p^k m with gcd(m, p)=1, effectively reducing the non-coprime case to the coprime classification. This provides a unified criterion for trivial cyclically covering subspaces across all n and highlights the role of the residue trace in translating modular structure to the field-trace conditions of Huang. The findings offer a complete classification tool and pave the way for further exploration of covering properties via modular representations and trace methods.

Abstract

A subspace of is called \textit{cyclically covering} if the whole space is the union of the cyclic shifts of . The case itself is the only covering subspace, is of particular interest. Recently, Huang solved this problem completely under the condition using primitive idempotents and trace functions, and explicitly posed the non-coprime case as an open question. This paper provides a complete answer to Huang's question. We prove that if where and , then if and only if . This result fully reduces the non-coprime case to the coprime case settled by Huang. Our proof employs the structure theory of cyclic group algebras in modular characteristic.
Paper Structure (17 sections, 9 theorems, 25 equations)

This paper contains 17 sections, 9 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. Background and Problem Statement
  3. The Coprime Case: Huang's Complete Solution
  4. Our Contribution
  5. Structure of the Group Algebra
  6. Preliminaries on Group Algebras
  7. Decomposition in Modular Characteristic
  8. Semisimple Decomposition of $\mathbb{F}_q H$
  9. The Residue Trace Map and Proof of the Main Theorem
  10. Motivation and Definition of the Residue Trace Map
  11. Properties of the Residue Trace Map
  12. Translating the Covering Condition to Algebraic Form
  13. Reduction to a Universal Element
  14. Computing the Key Product
  15. Connection to Huang's Trace Condition
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $\gcd(n, q)=1$. Then $h_q(n)=0$ if and only if for each $t = 0, 1, \ldots, r-1$, there exists a coset of $\langle \theta^{k_t} \rangle$ in the multiplicative group $\mathbb{F}_{q^{m_t}}^\times$ such that $\operatorname{Tr}_{\mathbb{F}_q}^{\mathbb{F}_{q^{m_t}}}(x) \neq 0$ for every $x$ in this co

Theorems & Definitions (20)

  • Theorem 1.1: Huang Huang2024, Theorem 3.7
  • Theorem 1.2: Main Theorem
  • Theorem 2.1
  • proof
  • Remark 2.2
  • Theorem 2.3
  • proof
  • Remark 2.4
  • Definition 3.1: Residue Trace Map
  • Lemma 3.2
  • ...and 10 more