### Smallest cyclically covering subspaces of F_q^n

Combinatorics Seminar

5th March 2019, 11:00 am – 12:00 pm

Howard House, 4th Floor Seminar Room

Let σ: F_q^n -> F_q^n be the cyclic shift operator; the map which permutes the entries of each vector by shifting them cyclically one step clockwise. We say a subspace U ≤ F_q^n is cyclically covering if the union of the cyclic shifts of U is equal to F_q^n, i.e. \bigcup_{r=0}^{n-1}σ^r(U) = F_q^n. This talk will investigate the problem of determining the minimum possible dimension of a cyclically covering subspace of F_q^n. This is a natural generalisation of a problem posed in 1991 by Peter Cameron who investigated the binary case. Using techniques from combinatorics, representation theory and the theory of finite fields we prove upper and lower bounds for each fixed q and answer the question completely for infinitely many values of q and n. Finally we consider the analogous problem for general representations of groups. This is joint work with Peter Cameron and David Ellis.

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