Summary: We introduce new notions of the extended basic constraint qualification and the strong basic constraint qualification and discuss their relationship with other fundamental concepts such as the basic constraint qualification and the metric regularity; in particular we provide a solution to an open problem of Lewis and Pang on characterizing the metric regularity in terms of normal cones. We present a characterization of error bounds for convex inequalities in terms of the strong basic constraint qualification. As applications, we study the linear regularity for infinite collections of closed convex sets in a Banach space.