{"ID":6497654,"CreatedAt":"2026-07-13T01:19:40.13847098Z","UpdatedAt":"2026-07-14T01:36:59.12045529Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2607.09478","arxiv_id":"2607.09478","title":"A divisibility theorem for odd $J$-characteristics of two-level designs","abstract":"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.","short_abstract":"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...","url_abs":"https://arxiv.org/abs/2607.09478","url_pdf":"https://arxiv.org/pdf/2607.09478v1","authors":"[\"Pieter Thijs Eendebak\"]","published":"2026-07-10T14:53:04Z","proceeding":"math.CO","tasks":"[\"math.CO\",\"math.ST\"]","methods":"[]","has_code":false}