  Quantum error correction and fault-tolerance make it possible to perform
quantum computations in the presence of imprecision and imperfections of
realistic devices. An important question is to find the noise rate at which
errors can be arbitrarily suppressed. By concatenating the 7-qubit Steane and
15-qubit Reed-Muller codes, the 105-qubit code enables a universal set of
fault-tolerant gates despite not all of them being transversal. Importantly,
the CNOT gate remains transversal in both codes, and as such has increased
error protection relative to the other single qubit logical gates. We show that
while the level-1 pseudo-threshold for the concatenated scheme is limited by
the logical Hadamard, the error suppression of the logical CNOT gates allows
for the asymptotic threshold to increase by orders of magnitude at higher
levels. We establish a lower bound of $1.28~\times~10^{-3}$ for the asymptotic
threshold of this code which is competitive with known concatenated models and
does not rely on ancillary magic state preparation for universal computation.
