arXiv:2608.01134 · v3
Perfect Codes As Exact Minimizers Of Quadratic Discrepancy In q-ary Hamming Spaces
摘要
Total quadratic ball discrepancy measures the deviation of codeword counts from uniformity over all Hamming balls. We prove that, whenever a Hamming space and cardinality admit a perfect code, the discrepancy minimizers among all subsets of that cardinality are precisely the perfect codes. This extends Barg's binary minimizing result and characterizes equality over arbitrary finite alphabets. At parameter sets that satisfy the sphere-packing condition and Lloyd's integral-root condition for correcting one or two errors, we give an explicit lower bound whose attainment is equivalent to perfect tiling: the discrepancy that a perfect code with these parameters would have. For one error, these include parameters at which the existence of a perfect code is open. For alphabets of size at least four, the discrepancy is an affine function of an energy with a strictly completely monotonic potential; for these alphabets, known universal energy bounds also give these lower bounds, and, when a perfect code exists, known universal optimality results also give the characterization of the minimizers. Over the ternary alphabet, complete monotonicity fails, and we prove the ternary bounds directly at all Hamming parameters and at the Golay parameters, the only nontrivial ternary cases.
作者备注：38 pages; no figures. Revised and expanded presentation, clarified hypotheses and supporting arguments, and updated references. Exact-arithmetic verification companion: https://doi.org/10.5281/zenodo.22084106