  The covering radius is a fundamental property of linear codes that
characterizes the trade-off between storage and access in linear data-query
protocols. The generalized covering radius was recently defined by Elimelech
and Schwartz for applications in joint-recovery of linear data-queries. In this
work we extend a known bound on the ordinary covering radius to the generalized
one for all codes satisfying the chain condition -- a known condition which is
satisfied by most known families of codes. Given a generator matrix of a
special form, we also provide an algorithm which finds codewords which cover
the input vectors within the distance specified by the bound. For the case of
Reed-Muller codes we provide efficient construction of such generator matrices,
therefore providing a faster alternative to a previous generalized covering
algorithm for Reed-Muller codes.
