A Time-Frequency Framework for GKP Codes

• URL: http://arxiv.org/abs/2609.10802v2
• Date: Fri, 11 Sep 2026 21:33:49 GMT
• Title: A Time-Frequency Framework for GKP Codes
• Abstract summary: We develop a time-frequency framework for lattice GKP codes in which ideal codewords are realized in the modulation space $Minfty$.<n>We prove that the normalized block map is an isometry, obtain an explicit recovering projection, and derive stable logical reconstruction.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We develop a time--frequency framework for lattice GKP codes in which ideal codewords are realized in the modulation space $M^\infty$ and identified, through a vector-valued Zak transform, with a finite logical fibre over the continuous syndrome torus. Multi-window Gabor analysis then represents the logical vector by a finite block of adjoint-lattice coefficients. We prove that the normalized block map is an isometry, obtain an explicit recovering projection, and derive stable logical reconstruction. We further construct normalizable GKP approximants as lattice-envelope Gabor multipliers and establish weak-$*$ convergence and asymptotically isometric encoding. Finally, we recover displacement syndromes from phase relations between translated coefficient blocks and quantify their stability under additive perturbations.

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