含4个与8个洞的规定差匹配：兼容边界的傅里叶滤波器
Prescribed-Difference Matchings with Four and Eight Holes: Fourier Filters for Compatible Boundaries
浏览论文内容
中文总结 AI 辅助
该研究在有限域F₂^s中证明，当s≥3时所有4-洞规定差匹配实例、s≥4时所有8-洞规定差匹配实例均存在解，通过带符号行列式编码与沃尔什变换完成补全验证，相关成果存档于Zenodo。
中文摘要 AI 辅助
设s≥2，且v₁,…,v_{2^{s-1}}∈F₂^s∖{0}的和为零。规定差匹配问题询问是否可将F₂^s划分为若干对，使得这些对的差（计重数）恰好为这些向量。固定超平面H≤F₂^s，h-洞实例指恰好h个规定差位于H中的实例（计重数）。我们证明，当s≥3时，每个4-洞实例都存在解；当s≥4时，每个8-洞实例都存在解。因此，除全局零和条件外，无需对重数或洞的和施加其他限制。在放置好内部对后，将它们的端点从H确定的两个仿射半空间中删除，剩余向量必须与规定的交叉差配对。局部有效的内部对放置未必能实现此类补全。我们通过带符号行列式的系数对可能的补全进行编码，并使用沃尔什变换在保持交叉轮廓固定的情况下对所有不相交的合法放置求和。该和非零则保证至少有一个放置可扩展为完整匹配。4-洞定理与零和8-洞定理已通过理论证明；一般8-洞证明则使用了三项与交叉轮廓无关的有限精确验证：局部滤波器计算、剩余8-洞构型的分类，以及针对这些构型的整数证书。验证软件、精确输入与记录输出均存档于版本化的Zenodo记录中。
英文摘要
Let $s\geq 2$ and let $v_1,\ldots,v_{2^{s-1}}\in F_2^s\setminus\{0\}$ have sum zero. The prescribed-difference matching problem asks whether $F_2^s$ can be partitioned into pairs whose differences, counted with multiplicity, are exactly these vectors. Fix a hyperplane $H\leq F_2^s$. An $h$-hole instance is one in which exactly $h$ prescribed differences lie in $H$, counted with multiplicity. We prove that every four-hole instance has a solution for $s\geq 3$ and every eight-hole instance has a solution for $s\geq 4$. Thus no restriction on the multiplicities or on the sum of the holes is needed beyond the global zero-sum condition. After the internal pairs are placed, their endpoints are deleted from the two affine halves determined by $H$, and the remaining vectors must be paired with the prescribed crossing differences. A locally valid placement of the internal pairs need not admit such a completion. We encode the possible completions by coefficients of signed determinants and use the Walsh transform to sum over all disjoint legal placements while keeping the crossing profile fixed. Nonvanishing of this sum guarantees that at least one placement extends to a full matching. The four-hole and zero-sum eight-hole theorems are proved theoretically. The general eight-hole proof uses three finite exact verifications, independent of the crossing profile: a local filter calculation, a classification of the remaining eight-hole configurations, and integer certificates for those configurations. The verification software, exact inputs, and recorded outputs are archived in a versioned Zenodo record.
