Optimal two-mode bosonic loss codes from finite group symmetry

Photon loss is a dominant noise process in bosonic quantum hardware, including superconducting cavities. Fixed-total-photon-number qubit encodings in two bosonic modes retain the loss-detection advantage of dual-rail qubits while supporting photon loss correction. Optimizing entanglement fidelity in this setting for 4 at most n at most25 over arbitrary encoders and decoders reveals finite-group structure in every best-found code: 11 correspond to two-dimensional irreducible representations and 11 to reducible ones. Motivated by this emergence, we derive the necessary-and-sufficient Knill-Laflamme conditions for arbitrary finite-group-invariant codes, reducing their construction to equations on representation multiplicity spaces. For every finite subgroup of SU(2) and every tensor rank, we explicitly construct the minimum number of operators required to impose all corresponding loss-correction constraints. These results produce analytic counterparts to most numerical codes and predict constructions missed by the initial search. Exact MacWilliams-Farkas certificates show that our codes achieve the maximum possible loss distance in 21 of the 22 sectors, and we prove that the resulting distance bound is monotone in total photon number. Beyond the scan, we identify a binary-polyhedral sequence with photon number n d= (3d 2+1)/4 and construct each corresponding code through distance d=10. To our knowledge, the constructed (n,d)=(28,6),(49,8),(76,10) codes give the smallest reported n for their respective distances; all members through d=9 attain the fixed- n LP distance bound. Together, these constructions, certificates, and a symmetry-reduced constraint count provide evidence for an infinite code family conjectured to attain every distance at the minimum photon number.