  Self-orthogonal $J$-affine variety codes have been successfully used to
obtain quantum stabilizer codes with excellent parameters. In a previous paper
we gave formulae for the dimension of this family of quantum codes, but no
bound for the minimum distance was given. In this work, we show how to derive
quantum stabilizer codes with designed minimum distance from $J$-affine variety
codes and their subfield-subcodes. Moreover, this allows us to obtain new
quantum codes, some of them either, with better parameters, or with larger
distances than the previously known codes.
