  Motivated by an application to database linear querying, such as private
information-retrieval protocols, we suggest a fundamental property of linear
codes -- the generalized covering radius. The generalized covering-radius
hierarchy of a linear code characterizes the trade-off between storage amount,
latency, and access complexity, in such database systems. Several equivalent
definitions are provided, showing this as a combinatorial, geometric, and
algebraic notion. We derive bounds on the code parameters in relation with the
generalized covering radii, study the effect of simple code operations, and
describe a connection with generalized Hamming weights.
