AI 中文总结
研究有限阿贝尔群中\((G,\cB_k^x)\)何时为2-设计这一问题，通过特征理论方法，证明\((G,\cB_k^x)\)是非平凡2-设计仅当\(G\)是初等阿贝尔\(p\)群，还给出子集和1-设计刻画及一般算术限制。
AI 中文摘要
有限阿贝尔群上的子集和处于加法组合学、设计理论和编码理论的交叉点。设\(G\)为有限阿贝尔群，\(\cB_k^x\)是\(G\)中元素和为\(x\in G\)的\(k\)子集族。本文研究关联结构\((G,\cB_k^x)\)何时是区组设计。Falcone和Pavone解决了初等阿贝尔\(p\)群的情况。Pavone进一步询问，对于任意有限阿贝尔群\(G\)，零和关联结构\((G,\cB_k^0)\)是否仅当\(G\)是初等阿贝尔\(p\)群时才能是非平凡2-设计。我们以更强的形式解决了这个开放性问题，即对于每个\(x\in G\)，\((G,\cB_k^x)\)仅当\(G\)是初等阿贝尔\(p\)群时才能是非平凡2-设计。证明发展了一种用于子集和设计的特征理论方法，利用区组上的特征和来约束特征群\(\widehat G\)的结构。该方法还给出了子集和1-设计的完整刻画以及对任意有限阿贝尔群上子集和设计的一般算术限制，扩展了之前已知的有限阿贝尔\(p\)群的相应结果。
英文摘要
Subset sums over finite abelian groups lie at the intersection of additive combinatorics, design theory, and coding theory. Let $G$ be a finite abelian group, and let $\cB_k^x$ be the family of $k$-subsets of $G$ whose elements sum to $x\in G$. This paper studies when the incidence structure $(G,\cB_k^x)$ is a block design. The elementary abelian $p$-group case was settled by Falcone and Pavone. Pavone (\emph{Des. Codes Cryptogr.} 91 (2023), 2585--2603) further asked whether, for an arbitrary finite abelian group $G$, the zero-sum incidence structure $(G,\cB_k^0)$ can be a nontrivial $2$-design only when $G$ is an elementary abelian $p$-group. We settle this open question in the stronger form that, for every $x\in G$, $(G,\cB_k^x)$ can be a nontrivial $2$-design only if $G$ is an elementary abelian $p$-group. The proof develops a character-theoretic approach to subset-sum designs, using character sums over the blocks to constrain the structure of the character group $\widehat G$. The approach also yields a complete characterization of subset-sum $1$-designs and general arithmetic restrictions on subset-sum designs over arbitrary finite abelian groups, extending the corresponding results previously known for finite abelian $p$-groups.
