Generating binary spaces

Let r and ρ be integers satisfying r ≥ ρ ≥ 3, and let F2r denote the elementary 2-group of rank r. We show that the maximum possible cardinality of a generating subset A ⊆ F2r, such that not all elements of F2r are representable as a sum of fewer than ρ elements of A, is (ρ + 1)2r-ρ. This proves a conjecture of Zemor and solves a well-known problem, related to covering radii of linear binary codes.Indeed, we give a full description of all those generating subsets A⊆F2r of cardinality |A|>(ρ + 5)2r-ρ-1 such that not all elements of F2r are representable as a sum of fewer than ρ elements of A.