A 67%-Rate CSS Code on the FCC Lattice: [[192,130,3]] from Weight-12 Stabilizers

We construct a three-dimensional Calderbank-Shor-Steane (CSS) stabilizer code on the Face-Centered Cubic (FCC) lattice. Physical qubits reside on the edges of the lattice \(coordination $K=12$\); X-stabilizers act on octahedral voids and Z-stabilizers on vertices, both with uniform weight 12. Computational verification confirms CSS validity \($H_{X}H_{Z}^{T}=0$ over GF(2\)) and reveals $k=2L^{3}+2$ logical qubits: $k=130$ at $L=4$ and $k=434$ at $L=6$, yielding encoding rates of 67.7% and 67.0% respectively. The minimum distance $d=3$ is proven exactly by exhaustive elimination of all weight-$\le 2$ candidates combined with constructive weight-3 non-stabilizer codewords. The code parameters are [[192, 130, 3]] at $L=4$ and [[648, 434, 3]] at $L=6$. This rate is 24x higher than the cubic 3D toric code \(2.8% at $d=4$\), though at a lower distance \($d=3$ vs. $d=4$\); the comparison is across different distances. The high rate originates in a structural surplus: the FCC lattice has $3L^{3}$ edges but only $L^{3}-2$ independent stabilizer constraints, leaving $k=2L^{3}+2$ logical degrees of freedom. We provide a minimum-weight perfect matching (MWPM) decoder adapted to the FCC geometry, demonstrate a 10x coding gain at $p=0.001$ \(and 63x at $p=0.0005$\), and discuss implications for fault-tolerant quantum computing on neutral-atom and photonic platforms.