Cyclically covering subspaces in F<sub>2</sub><sup>n</sup>

Journal article (2021)

Authors

James Aaronson University of Oxford
C.E. Groenland University of Oxford
Tom Johnston University of Oxford
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Cyclic shift Cyclically covering subspaces Isbell's conjecture Smallest codimension
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Published Date

2021

Language

English

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Abstract

A subspace of F2n is called cyclically covering if every vector in F2n has a cyclic shift which is inside the subspace. Let h2(n) denote the largest possible codimension of a cyclically covering subspace of F2n. We show that h2(p)=2 for every prime p such that 2 is a primitive root modulo p, which, assuming Artin's conjecture, answers a question of Peter Cameron from 1991. We also prove various bounds on h2(ab) depending on h2(a) and h2(b) and extend some of our results to a more general set-up proposed by Cameron, Ellis and Raynaud.