These CSS-like codes are defined simply from two integer matrices. We can encode qudit, modes, or rotors. Codes can have finite or infinite Fock-state support.
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Codewords are projected coherent states, unifying many different previously known constructions (pair-cat, two-mode binomial, dual-rail, SU(N) coherent, various repetition codes, some chi-squared codes) while also defining codes from lattices and algebraic varieties.
Error correction is relatively simple: losses move states into a different Fock subspace, and measuring particular linear combinations of photon numbers determines where the state is. Tracking without correcting is possible with infinite-support Fock-state codes.
Codewords normalization and overlap are governed by a Gelfand-Kapranov-Zelevinsky (GKZ) hypergeometric function, and we anticipate interesting connections to this deep literature down the road.
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Yijia Xu, Yixu Wang, and integer-homology expert Christophe Vuillot!