Perfect codes as exact minimizers of quadratic discrepancy in q-ary Hamming spaces

What the math gives to ML

The paper supplies an exact global objective for measuring how uniformly a finite set of q-ary codewords covers all Hamming balls, with perfect codes attaining the minimum whenever they exist. This is transferable as a structured regularizer for discrete representation systems such as VQ tokenizers, product quantizers, and error-robust codebooks: instead of only separating codewords, it encourages every region of the discrete Hamming space to receive the correct mass at multiple radii. The exact theorem applies to hard fixed-cardinality subsets, while neural use requires a differentiable soft-assignment relaxation and should be evaluated for robustness and codebook utilization rather than claiming perfect-code optimality.