  Topological subsystem codes in three spatial dimensions allow for quantum
error correction with no time overhead, even in the presence of measurement
noise. The physical origins of this single-shot property remain elusive, in
part due to the scarcity of known models. To address this challenge, we provide
a systematic construction of a class of topological subsystem codes in three
dimensions built from abelian quantum double models in one fewer dimension. Our
construction not only generalizes the recently introduced subsystem toric code
[Kubica and Vasmer, Nat. Commun. 13, 6272 (2022)] but also provides a new
perspective on several aspects of the original model, including the origin of
the Gauss law for gauge flux, and boundary conditions for the code family. We
then numerically study the performance of the first few codes in this class
against phenomenological noise to verify their single-shot property. Lastly, we
discuss Hamiltonians naturally associated with these codes, and argue that they
may be gapless.
