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. , 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.