AI 中文总结
研究二级设计奇数 \(J\)-特征的整除定理，当因子数 \(n\) 为奇数且特定子集 \(J\)-特征为零时，最高阶 \(J\)-特征可被 \(2^{n - 1}\) 整除，解决了关于特定强度设计运行次数的猜想，整除界精确且由偶权重半分数达到。
AI 中文摘要
我们证明了关于二级设计带符号 \(J\)-特征的一个整除定理：若因子数 \(n\) 为奇数且因子的每个适当奇基数子集的 \(J\)-特征都为零，则最高阶 \(J\)-特征可被 \(2^{n - 1}\) 整除。作为一个算术结果，任何二级设计，其 \(J\)-特征在一阶、二阶、三阶、五阶和七阶为零，但具有非零奇数阶 \(J\)-特征，则必须至少有 \(256\) 次运行。这统一解决了Eendebak、Schoen、Vazquez和Goos（2023）关于不存在具有 \(56\) 或 \(64\) 次运行的某些强度为三的偶奇设计的猜想。该整除界在每个奇数阶都是精确的，并且由偶权重半分数达到。
英文摘要
We prove a divisibility theorem for the signed $J$-characteristics of two-level designs: if the number of factors $n$ is odd and every $J$-characteristic of a proper odd-cardinality subset of factors vanishes, then the top $J$-characteristic is divisible by $2^{n-1}$. As an arithmetic consequence, any two-level design whose $J$-characteristics vanish in orders one, two, three, five, and seven but which has a nonzero odd-order $J$-characteristic must have at least $256$ runs. This settles, uniformly in the number of factors, a conjecture of Eendebak, Schoen, Vazquez, and Goos (2023) on the nonexistence of certain strength-three even--odd designs with $56$ or $64$ runs. The divisibility bound is sharp at every odd order and is attained by the even-weight half-fraction.
