Quantum 9, 1826 (2025). https://doi.org/10.22331/q-2025-08-08-1826 We introduce a new class of qubit codes that we call Evenbly codes, building on a previous proposal of hyperinvariant tensor networks. Its tensor network description consists of local, non-perfect tensors describing CSS codes interspersed with Hadamard gates, placed on a hyperbolic $\\{p,q\\}$ geometry with even $q\geq 4$, yielding an infinitely large class of subsystem codes. We construct an example for a $\\{5,4\\}$ manifold and describe strategies of logical gauge fixing that lead to different rates $k/n$ and distances $d$, which we calculate analytically, finding distances which range from $d=2$ to $d \sim n^{2/3}$. Investigating threshold performance under erasure, depolarizing, and pure Pauli noise channels, we find that the code exhibits a depolarizing noise threshold of about 19.1% in the code-capacity model and 50% for pure Pauli and erasure channels under suitable gauges. We also test a constant-rate version with $k/n = 0.125$, finding excellent error resilience (about 40%) under the erasure channel. Recovery rates for these and other settings are studied both under an optimal decoder as well as a more efficient but non-optimal greedy decoder. We also consider generalizations beyond the CSS tensor construction, compute error rates and thresholds for other hyperbolic geometries, and discuss the relationship to holographic bulk/boundary dualities. Our work indicates that Evenbly codes may show promise for practical quantum computing applications.