Symplectic Barnes-Wall GKP Codes: Deterministic $O(N \log^2 N)$ Decoding and Logarithmic Rate Scaling

We construct an explicit symplectic realization of the Barnes-Wall lattice that yields a family of multimode Gottesman-Kitaev-Preskill (GKP) codes with encoding rate $R = \frac{1}{2}\log_2 N$ and a deterministic $O(N\log^2 N)$ bounded-distance decoder. The recursive generator $G_{m+1} = \bigl(\begin{smallmatrix} G_m & 0 \\ G_m & R_m G_m \end{smallmatrix}\bigr)$ with $R_m = I + \Omega$ simultaneously guarantees symplectic integrality for valid quantum stabilizers and preserves the exact Barnes-Wall decoding structure through a chain of isometric isomorphisms. The code distance is constant at $\Delta^2 = 1$ (in units of $2\pi$), representing an explicit distance--rate tradeoff in which logarithmic encoding efficiency is achieved at the cost of non-scaling protection. This construction provides a deterministic, space-efficient paradigm for GKP error correction in platforms supporting non-local modular connectivity.