The Quantum Hamming Bound in Arbitrary Local Dimension

Abstract:The quantum Hamming bound arises from a direct dimension estimate for exact quantum error-correcting codes. For nondegenerate codes, distinct correctable errors map the code space into orthogonal subspaces, so dimension counting directly gives the bound; degeneracy allows these subspaces to overlap and invalidates the argument. Although linear-programming, asymptotic, and structural approaches have advanced the problem from several directions, the general local-dimension case has remained open. Here we prove the quantum Hamming bound for every local dimension $q\ge3$. Building on the exact two-center Fourier reduction, we develop new overlap estimates for nonbinary Hamming spaces. For $q\ge4$, Jacobi estimates and monotonicity give a uniform gap; qutrits form the boundary case where the gap vanishes and are handled by a filtered correction. This completes the finite-length nonbinary case; together with the existing qubit result, it establishes the quantum Hamming bound for arbitrary local dimension.

Main 6 pages+ SM 39 pages. Revised version, adding 2 figures

arXiv:2606.22538 [quant-ph]

(or arXiv:2606.22538v2 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2606.22538 Focus to learn more arXiv-issued DOI via DataCite

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