汉明空间中基于二次偏差的量化铺砌稳定性
Quantitative tiling stability from quadratic discrepancy in Hamming spaces
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中文总结 AI 辅助
该研究在汉明空间中证明了二次球偏差相关的铺砌缺陷稳定性，推导了不同参数下的系数，为非平凡完美码提供了稳定性保证，关联了偏差与球噪声的多种均匀性度量。
中文摘要 AI 辅助
二次球偏差在有限汉明空间的码上定义了一种能量。在完美码参数下，其精确极小值点为完美码。我们固定字母大小、长度和码的基数，对所有具有这些参数的码进行比较。我们证明了铺砌缺陷稳定性：超过完美码基准的偏差控制了特征球覆盖重数与1的平方偏差。对于满足球填充和劳埃德积分性的单错误参数，下界系数为κ_{n,q}/q²≥1。均匀下限1是紧的，而经认证的依赖参数的系数可大得多，同时也存在显式的依赖参数的上界估计，且两个经认证的系数可能相差甚远。对于满足球填充可分性且在汉明重量范围内有两个不同整数劳埃德根的任意n≥5的双错误参数对，我们在不假设存在完美码的情况下得到了显式正系数。对于大小至少为4的字母，该条件系数具有闭式表达式和固定字母渐近性。重复码和戈莱码族的直接证明，结合完美码分类，为每个非平凡完美码提供了铺砌缺陷稳定性。此处的稳定性涉及球覆盖重数轮廓，而非与特定完美码的对称差接近程度。该缺陷也是均匀球噪声下的归一化卡方平滑误差，因此超出的偏差控制了球噪声输出均匀性的空洞、重叠、缺陷环境点、全变差及雷尼散度。竞争码不必是线性码，也不必满足距离或纠错约束。
英文摘要
Quadratic ball discrepancy measures how uniformly a code meets Hamming balls over all centers and radii. Perfect codes minimize this quantity among codes of the same cardinality. We prove a quantitative refinement: at the parameters of every nontrivial perfect code, excess discrepancy is at least a positive multiple of the discrepancy at the correction radius. The latter equals the squared deviation of the covering multiplicity from one, divided by the square of the code cardinality. The comparison applies to all codes of that cardinality. For one-error parameters, the coefficient is at least one, which is sharp uniformly over the parameters; explicit coefficients depending on the parameters can be much larger. For two-error parameters of length at least five, we obtain a positive coefficient against an algebraic benchmark under the sphere-packing and Lloyd integrality conditions, even when existence of a perfect code is unknown. The multiplicity defect also measures the failure of uniform ball noise to smooth a code distribution. Consequently, the same discrepancy excess gives explicit bounds on holes, overlaps, and divergence from the uniform distribution.
发表机构
- Tel-Hai University of Kiryat Shmona and the Galilee（泰尔海基里亚特谢莫纳和加利利大学）
- MIGAL – Galilee Research Institute（MIGAL-加利利研究所）
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