Cyclically covering subspaces

Carla Groenland, University of Oxford
Fine Hall 224

A subspace of F_2^n is called "cyclically covering" if every vector in F_2^n has a cyclic shift which is inside the subspace. Let h_2(n) denote the largest possible codimension of a cyclically covering subspace of F_2^n. We show that h_2(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.  In this talk, I will try to explain how we reduce the problem to a problem on finding odd subgraphs in which each vertex has odd outdegree in directed Cayley graphs, how additive combinatorics comes to the rescue and which open problems I would like to see solved.

This is joint work with James Aaronson and Tom Johnston.