发表机构
MIGAL–Galilee Research Institute; Tel-Hai University of Kiryat Shmona and the Galilee（MIGAL-加利利研究所; 基里亚特谢莫纳和加利利泰尔海大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究证明q元汉明空间中，当固定长度和基数存在完备码时，完备码是二次偏差的精确极小化元，其存在性等价于显式变分目标的达成。
AI 中文摘要
斯托拉斯基不变原理将二次偏差转化为能量极小化问题，巴尔针对二元汉明空间发展了该原理的形式，并证明二元完备码在相同长度和基数的二元码中使总二次球偏差最小。我们证明了精确的有限字母对应结论：当固定汉明空间和基数存在完备码时，完备码恰好是偏差极小化元，竞争对象为规定基数的任意子集，无线性性或最小距离假设。更明确地，我们给出存在性已知前的仅含参数的显式下界基准：对每个算术可容许的单错误参数集，以及每个满足球填充和积分劳埃德根条件的非平凡双错误参数集，对应基准中的等式等价于完备铺砌。这些界通过显式傅里叶-克劳奇克和多项式证书证明，而非求解数值线性规划。因此，在所有覆盖的参数范围内，完备码的存在性等价于显式变分目标的达成。
英文摘要
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 the necessary parameter sets for correcting one or two errors, we give explicit lower bounds whose attainment is equivalent to perfect tiling, even when existence is unresolved. For alphabets of size at least four, the numerical values follow from known universal energy bounds; we establish their discrepancy normalization and the attainment criterion. The ternary discrepancy potential falls outside the completely monotonic regime of those bounds, and separate exact certificates settle the ternary Hamming and Golay cases.
Comments32 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
