Cyclically covering subspaces in F2n

James Aaronson; Carla Groenland; Tom Johnston

doi:10.1016/j.jcta.2021.105436

Cyclically covering subspaces in F₂ⁿ

James Aaronson, Carla Groenland^*, Tom Johnston

^*Corresponding author for this work

Research output: Contribution to journal › Article › Scientific › peer-review

2 Citations (Scopus)

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.

Original language	English
Article number	105436
Journal	Journal of Combinatorial Theory. Series A
Volume	181
DOIs	https://doi.org/10.1016/j.jcta.2021.105436
Publication status	Published - 2021
Externally published	Yes

Keywords

Cyclic shift
Cyclically covering subspaces
Isbell's conjecture
Smallest codimension

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10.1016/j.jcta.2021.105436

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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.",

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AU - Groenland, Carla

AU - Johnston, Tom

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AB - 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.

KW - Cyclic shift

KW - Cyclically covering subspaces

KW - Isbell's conjecture

KW - Smallest codimension

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SN - 0097-3165

VL - 181

JO - Journal of Combinatorial Theory. Series A

JF - Journal of Combinatorial Theory. Series A

M1 - 105436

ER -

Cyclically covering subspaces in F₂ⁿ

Abstract

Keywords

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