  Attaining fault tolerance while maintaining low overhead is one of the main
challenges in a practical implementation of quantum circuits. One major
technique that can overcome this problem is the flag technique, in which
high-weight errors arising from a few faults can be detected by a few ancillas
and distinguished using subsequent syndrome measurements. The technique can be
further improved using the fact that for some families of codes, errors of any
weight are logically equivalent if they have the same syndrome and weight
parity, as previously shown in [Phys. Rev. A 104, 042410 (2021)]. In this work,
we develop a notion of distinguishable fault set which captures both concepts
of flags and weight parities, and extend the use of weight parities in error
correction from [Phys. Rev. A 104, 042410 (2021)] to families of capped and
recursive capped color codes. We also develop fault-tolerant protocols for
error correction, measurement, state preparation, and logical T gate
implementation via code switching, which are sufficient for performing
fault-tolerant Clifford computation on a capped color code, and performing
fault-tolerant universal quantum computation on a recursive capped color code.
Our protocols for a capped or a recursive capped color code of any distance
require only 2 ancillas, assuming that the ancillas can be reused. The concept
of distinguishable fault set also leads to a generalization of the definitions
of fault-tolerant gadgets proposed by Aliferis, Gottesman, and Preskill.
