  We investigate a family of fault-tolerant quantum error correction schemes
based on the concatenation of small error detection or error correction codes
with the three-dimensional cluster state. We propose fault-tolerant state
preparation and decoding schemes that effectively convert every circuit-level
error into an erasure error, leveraging the cluster state's high threshold
against such errors. We find a set of codes for which such a conversion is
possible, and study their performance against the standard circuit-level
depolarizing model. Our best performing scheme, which is based on a
concatenation with a classical code, improves the threshold by $16.5\%$ and
decreases the spacetime overhead by $32\%$ compared to the scheme without
concatenation, with each scheme subject to a physical error rate of $10^{-3}$
and achieving a logical error rate of $10^{-6}$.
