Symplectic Barnes-Wall GKP Codes: Deterministic O\(N log² N\) 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}log₂ N and a deterministic O\(Nlog² N\) 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 Δ² = 1 in units of $2pi$, 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.