Smallest cyclically covering subspaces of $\mathbb{F}_q^n$, and lower bounds in Isbell's conjecture

TL;DR

This paper studies the minimal dimension of cyclically covering subspaces in finite fields, providing bounds, exact results for infinitely many cases, and implications for Isbell's conjecture using combinatorics, representation theory, and finite field theory.

Contribution

It offers new bounds and exact solutions for the minimal dimension of cyclically covering subspaces, advancing understanding of Isbell's conjecture and its generalizations.

Findings

Established bounds for cyclically covering subspaces.

Solved the problem for infinitely many n values for each fixed q.

Provided lower bounds relevant to Isbell's conjecture.

Abstract

For a prime power $q$ and a positive integer $n$, we say a subspace $U$ of ${\mathbb{F}_q^n}$ is {\em cyclically covering} if the union of the cyclic shifts of $U$ is equal to $\mathbb{F}_q^n$. We investigate the problem of determining the minimum possible dimension of a cyclically covering subspace of $\mathbb{F}_q^n$. (This is a natural generalisation of a problem posed in 1991 by the first author.) We prove several upper and lower bounds, and for each fixed $q$, we answer the question completely for infinitely many values of $n$ (which take the form of certain geometric series). Our results imply lower bounds for a well-known conjecture of Isbell, and a generalisation theoreof, supplementing lower bounds due to Spiga. We also consider the analogous problem for general representations of groups. We use arguments from combinatorics, representation theory and finite field theory.