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.
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