Given a prime p and positive integer n, let fp(n) denote the minimal number of hyperplanes in an irredundant covering of Fnp such that the normal vectors of the hyperlanes span the whole space. The function fp(n) appears to be in connection to several longstanding conjectures in linear algebra and group theory, such as the Alon-Jaeger-Tarsi conjecture, the Additive Basis conjecture, and a conjecture of Pyber on irredundant coset covers of abelian groups. We prove that log p fp(n) = Omega log log p . n , and use this result to make substantial progress on each of the aforementioned conjectures.