Smallest cyclically covering subspaces of Fqn, and lower bounds in Isbell's conjecture
Date
10/2019Metadata
Show full item recordAbstract
For a prime power q and a positive integer n, we say a subspace U of Fqn is cyclically covering if the union of the cyclic shifts of U is equal to Fqn. We investigate the problem of determining the minimum possible dimension of a cyclically covering subspace of Fqn. (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.
Citation
Cameron , P J , Ellis , D & Raynaud , W 2019 , ' Smallest cyclically covering subspaces of F q n , and lower bounds in Isbell's conjecture ' , European Journal of Combinatorics , vol. 81 , pp. 242-255 . https://doi.org/10.1016/j.ejc.2019.06.004
Publication
European Journal of Combinatorics
Status
Peer reviewed
ISSN
0195-6698Type
Journal article
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