发表机构
South China Normal University; South China University of Technology（华南师范大学; 华南理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对有限阿贝尔群中的符号差集，建立了新的非存在性准则与存在性构造，排除了大量开放案例并解决了部分开放问题。
AI 中文摘要
符号差集（SDS）通过允许负系数扩展了普通差集。我们在有限阿贝尔群中建立了SDS的新非存在性准则和存在性构造。利用经典的自共轭素数方法和支持细化商方法，我们得到四个阻碍，包括C₂-商、近全支持和小缺陷阻碍。将这些准则累计应用于数据库中9≤v≤499的67823个特定群开放案例，排除了45361个案例。对于存在性，我们由来自Desarguesian spreads的PCP型正则部分差集构造了一个无限族，并利用四次乘法特征在C₅³中构造了一个显式的(125,28,3)-SDS，这些构造解决了另外三个开放案例。
英文摘要
Signed difference sets (SDSs) extend ordinary difference sets by allowing negative coefficients. We establish new nonexistence criteria and existence constructions for SDSs in finite abelian groups. Using the classical self-conjugate-prime method and a support-refined quotient method, we derive four obstructions, including the $C_2$-quotient, near-full-support, and small-defect obstructions. Applied cumulatively to the $67{,}823$ open group-specific cases in the database with $9\leq v\leq499$, these criteria rule out $45{,}361$ cases. For existence, we construct an infinite family from PCP-type regular partial difference sets arising from Desarguesian spreads and an explicit $(125,28,3)$-SDS in $C_5^3$ using quartic multiplicative characters. These constructions settle three further open cases.
