L = 3 mod 8 时 32-模 Hadamard 矩阵的显式半翻转族：结构放置与模塔分析

中文摘要 AI 辅助

受 Eliahou 在最小未解 Hadamard 阶 n=668 处的 64-模 Hadamard 构造启发，我们引入了在阶 n=4L（L ≡ 3 (mod 8)）处的 32-模 Hadamard 矩阵的显式半翻转族。一个主恒等式将半翻转假设 (s, s*, sq, (sq)*) 下的四序列 Golay 四元组条件简化为单序列类型受限自相关 c_k^tau(s)。该构造给出了 c_k^tau 的闭式表达式，在 L=11 处得到真正的 Hadamard 矩阵，并在每个 L ≡ 3 (mod 8) 处得到 32-模矩阵，包括未解阶 n=716 和 n=1132。L ≡ 3 (mod 4) 处 32-模的存在性归功于 Eliahou-Kervaire (2001)；Eliahou 在 J. Algebraic Combin. 2026 年的后续工作通过相同的假设和我们称为主恒等式的相关恒等式，在 L ≡ 3 (mod 16) 和 L ≡ 7 (mod 32) 处给出 64-模矩阵，涵盖了我们在该剩余类子类上的构造。因此，我们的贡献主要是结构性的。应用 Barrera Acevedo-O Cathain-Dietrich (2019) 和 Alvarez 等人 (2020) 的结果，该族在 YES 集中的每个素数 L ∈ {11, 19, 59} 上对任何群都是非余循环的，但在每个 L 上对 Goethals-Seidel Moufang 环 GS_{4L} 是伪余循环的。一个半翻转 H-集分解定理将任意两个族元素的对称差参数化为单个序列翻转集，赋予该族长度为 L 的 Hamming 立方体结构。一个符号模塔验证器（模 8 是 F_2-线性的）通过 k=14（L ≤ 115）对族中的真正 Hadamard 矩阵进行分类：YES 集经验上以 k=7 为界，反驳了四个 H4 预测，并在假设内排除了 L ∈ {179, 283}。L=11 处的 Grobner 基展示了一个先前未记录的偶 T0 块线性恒等式。所有代码和 JSON 证书：此 http URL

英文摘要

Motivated by Eliahou's 64-modular Hadamard construction at the smallest open Hadamard order n=668, we introduce an explicit half-flip family of 32-modular Hadamard matrices at orders n=4L for L = 3 (mod 8). A master identity reduces the four-sequence Golay-quadruple condition under the half-flip ansatz (s, s*, sq, (sq)*) to a single-sequence type-restricted autocorrelation c_k^tau(s). The construction yields a closed-form expression for c_k^tau, a true Hadamard matrix at L=11, and 32-modular matrices at every L = 3 (mod 8), including the open orders n=716 and n=1132. Existence of 32-modular at L = 3 (mod 4) is due to Eliahou-Kervaire (2001); Eliahou's 2026 follow-up in J. Algebraic Combin. gives 64-modular matrices at L = 3 (mod 16) and L = 7 (mod 32) via the same ansatz and correlation identity we call the master identity, subsuming our construction on that residue subclass. Our contribution is therefore primarily structural. Applying Barrera Acevedo-O Cathain-Dietrich (2019) and Alvarez et al. (2020), the family is non-cocyclic over any group at every prime L in {11, 19, 59} in the YES set, yet pseudococyclic over the Goethals-Seidel Moufang loop GS_{4L} at every L. A half-flip H-set decomposition theorem parameterizes the symmetric difference of any two family elements by a single sequence flip set, giving the family the structure of a length-L Hamming cube. A symbolic mod-tower verifier (mod-8 is F_2-linear) classifies true Hadamards in the family through k=14 (L <= 115): the YES set is empirically bounded by k=7, refuting four H4 predictions and excluding L in {179, 283} within the ansatz. A Grobner basis at L=11 exhibits a previously unrecorded even-T0-block linear identity. All code and JSON certificates: github.com/michelkulhandjian/hadamard-halfflip-structural

发表机构

• Rice University（莱斯大学）

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