Smallest cyclically covering subspaces of ${\mathbb{F}}_q^n$, and lower bounds in 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.

arXiv commentA brief discussion of the sharpness of the results in Section 4 has been added (along with some minor corrections and expositional changes), following helpful comments and suggestions from two anonymous referees. 13 pages

References in corpus (1)

• Cyclically covering subspaces in ${\mathbb{F}}_2^n$

Cited by in corpus (1)

• Cyclically covering subspaces in ${\mathbb{F}}_2^n$