AI 中文总结
本文建立环上Hadamard矩阵余循环发展的形式化框架，提出环余循环Hadamard矩阵概念，并证明存在四个24阶和八个28阶新的严格非群余循环的Hadamard等价类。
AI 中文摘要
本文基于一种带有结合性障碍的新上同调理论，为环上Hadamard矩阵的余循环发展建立了形式化框架。在此背景下，我们形式化了环余循环Hadamard矩阵的概念，使得经典的群上余循环Hadamard矩阵以及最近引入的环上伪余循环Hadamard矩阵都自然地嵌入到这个更广泛的代数架构中。为了验证该框架的计算可行性，我们证明了存在四个新的24阶Hadamard等价类和八个新的28阶Hadamard等价类，这些类在任何有限群上严格非余循环，且唯一地作为Moufang环和右Bol环上的环余循环发展而出现。
英文摘要
This paper establishes a formal framework for the cocyclic development of Hadamard matrices over loops based on a novel cohomology theory with associativity obstructions. In this context, we formalize the notion of a loop-cocyclic Hadamard matrix, so that both the classical cocyclic Hadamard matrices over groups and the recently introduced pseudococyclic Hadamard matrices over loops are naturally embedded within this broader algebraic architecture. To validate the computational viability of this framework, we prove the existence of four new Hadamard equivalence classes of order 24 and eight new classes of order 28 that are strictly non-cocyclic over any finite group, arising uniquely as loop-cocyclic developments over Moufang and right Bol loops.
Comments18 pages, revised version
