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

Abstract

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 $G_{m+1}=\left(\begin{array}{cc}{G}_m& 0\\ G_m& R_m{G}_m\end{array}\right)$ with $R_m=I+\hat{I}\text{c◯}$ 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 $\hat{I}{”}^2=1$ (in units of $2\ddot{I}€$), 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.