Researchers have demonstrated the validity of the quantum Hamming bound for exact quantum error-correcting codes in arbitrary local dimensions. This bound states that the code dimension multiplied by the number of correctable local error patterns must fit within the ambient Hilbert space. The main difficulty in proving this bound lay in degeneracy, where distinct physical errors can coincide on the code subspace, complicating the counting of disjoint error spheres.

The study addresses the problem for the non-binary range, starting at $Q=9$ (where $Q=q^2$ and $q$ is the local dimension). The proof extends to all $q \ge 3$. For $q \ge 4$, reduction to a Lloyd-square linear program with $Q \ge 16$ and critical length $n=4t+1$ shows a uniform "half-gap." The case of qutrits ($q=3$), which represents the boundary, required specific treatment using a quadratic-filtered Lloyd-square, an exact coefficient-certificate reduction, and a Stein-tangent positivity argument, as the "half-gap" disappears.

The demonstration confirms that, although degeneracy can merge error sectors, it is not enough to beat the Hamming count. This result, together with the independent theorem for the binary endpoint ($Q=4$), establishes the quantum Hamming bound in arbitrary local dimensions. This is crucial for the development of robust quantum error-correcting codes, a fundamental component for building fault-tolerant quantum computers.