  Recently we proposed a family of magic state distillation protocols that
obtains asymptotic performance that is conjectured to be optimal. This family
depends upon several codes, called "inner codes" and "outer codes." We
presented some small examples of these codes as well as an analysis of codes in
the asymptotic limit. Here, we analyze such protocols in an intermediate size
regime, using hundreds to thousands of qubits. We use BCH inner codes, combined
with various outer codes. We extend our protocols by adding error correction in
some cases. We present a variety of protocols in various input error regimes;
in many cases these protocols require significantly fewer input magic states to
obtain a given output error than previous protocols.
