发表机构
Gifu University（岐阜大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究有限域中由分圆类构成的外部差族，证明素数$p=12k+1$时三块族$\{C_0^6,C_2^6,C_4^6\}$构成EDF当且仅当$k$为平方数，并恢复了两块情形的经典刻画。
AI 中文摘要
我们从固定块数的角度研究有限域中由分圆类产生的外部差族。对于由偶数指标分圆类组成的族，EDF条件可以用分圆数之间的关系来表达。我们首先研究两块情形，并恢复了用二次型表示的经典刻画。我们的主要结果表明，对于素数$p=12k+1$，族$\{C_0^6,C_2^6,C_4^6\}$在$\mathbb{F}_p$中构成EDF当且仅当$k$是平方数。证明结合了分圆数的对称关系及其显式求值。
英文摘要
We study external difference families arising from cyclotomic classes in finite fields from the viewpoint of a fixed number of blocks. For families consisting of even-indexed cyclotomic classes, the EDF condition can be expressed in terms of relations among cyclotomic numbers. We first study the two-block case and recover a classical characterization in terms of quadratic forms. Our main result shows that, for a prime $p=12k+1$, the family $\{C_0^6,C_2^6,C_4^6\}$ forms an EDF in $\mathbb{F}_p$ if and only if $k$ is a square. The proofs combine symmetry relations of cyclotomic numbers with their explicit evaluations.
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