  In this paper we introduce the $q$-ratio block constrained minimal singular
values (BCMSV) as a new measure of measurement matrix in compressive sensing of
block sparse/compressive signals and present an algorithm for computing this
new measure. Both the mixed $\ell_2/\ell_q$ and the mixed $\ell_2/\ell_1$ norms
of the reconstruction errors for stable and robust recovery using block Basis
Pursuit (BBP), the block Dantzig selector (BDS) and the group lasso in terms of
the $q$-ratio BCMSV are investigated. We establish a sufficient condition based
on the $q$-ratio block sparsity for the exact recovery from the noise free BBP
and developed a convex-concave procedure to solve the corresponding non-convex
problem in the condition. Furthermore, we prove that for sub-Gaussian random
matrices, the $q$-ratio BCMSV is bounded away from zero with high probability
when the number of measurements is reasonably large. Numerical experiments are
implemented to illustrate the theoretical results. In addition, we demonstrate
that the $q$-ratio BCMSV based error bounds are tighter than the block
restricted isotropic constant based bounds.
