Abstract: Continuous-variable cat codes are encodings into a single photonic or phononic mode that offer a promising avenue for hardware-efficient fault-tolerant quantum computation. Protecting information in a cat code requires measuring the mode’s occupation modulo two, but this can be relaxed to a linear occupation-number constraint using the alternative two-mode pair-cat encoding. We construct multi-mode codes with similar linear constraints using any two integer matrices, 𝐻𝑋 and 𝐻𝑍 , satisfying the homological constraint of a quantum rotor code. Stabilizing operators are either linear combinations of occupation-number operators defined by rows of 𝐻𝑍 , or products of annihilation/creation operators whose powers form rows of 𝐻𝑋 . Just like the pair-cat code, syndrome extraction can be performed in tandem for both types of stabilizers using current superconducting-circuit designs. Our framework encompasses two-component cat, pair-cat, two-mode binomial, and aspects of chi-squared encodings while also yielding bosonic codes from homological products, lattices, and algebraic varieties. We cover several examples, including a surface-like code that is not obtained by concatenating a bosonic code with the surface code.