多维循环余循环Hadamard矩阵的计算框架
A computational framework for multidimensional loop-cocyclic Hadamard matrices
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- Universidad de Sevilla（塞维利亚大学）
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中文总结 AI 辅助
本文提出一个计算框架，将多维Hadamard矩阵的余循环构造从群推广到拟群，通过$\delta$-相容对简化搜索空间，并发现非结合拟群能产生群无法得到的三维Hadamard类。
中文摘要 AI 辅助
Hadamard矩阵的余循环发展最近已通过一种包含结合性障碍的上同调理论从群扩展到拟群。在有限域$\mathbb F_2$上，所得的循环余循环可以作为齐次线性系统的解来计算，使得该框架适用于精确计算。在本文中，我们研究了由有限拟群产生的多维Hadamard矩阵。我们引入了拟群$L$上$2$-余链的$\delta$-相容对的概念，作为通常的$2$-余循环恒等式的自然扩展，它允许使用两个不同的$2$-余链。它们构成一个向量空间，自然同构于$2$-余循环空间与$1$-余链空间的直积。这产生了从群到任意拟群的多维Hadamard矩阵的经典余循环构造的推广，同时保留了余循环方法的计算优势。所获得的分解表明，Hadamard等价类的确定可以从$\delta$-相容对的整个空间简化为普通$2$-余循环的较小空间，从而从搜索空间中消除了$2^n$的冗余因子。对所有阶为$4$和$8$的群以及一个阶为$8$的非结合拟群的计算结果表明，所提出的三维构造细化了经典的余循环Hadamard分类。虽然所有例子在二维中坍缩为单个等价类，但它们在三维中分裂为几个不同的类。特别是，非结合拟群产生了一个三维Hadamard类，该类不来自所考虑的任何群，表明多维构造检测到了在余循环矩阵级别不可见的结构信息。
英文摘要
The cocyclic development of Hadamard matrices has recently been extended from groups to loops by means of a cohomology theory that incorporates associativity obstructions. Over the finite field $\mathbb F_2$, the resulting loop-cocycles can be computed as solutions of homogeneous linear systems, making the framework suitable for exact computation. In this paper, we investigate multidimensional Hadamard matrices arising from finite loops. We introduce the notion of $δ$-compatible pairs of $2$-cochains over a loop $L$ as a natural extension of the usual $2$-cocycle identity, which allows the use of two distinct $2$-cochains. They form a vector space naturally isomorphic to the direct product of the space of $2$-cocycles and the space of $1$-cochains. This yields a generalization of the classical cocyclic construction of multidimensional Hadamard matrices from groups to arbitrary loops, while preserving the computational advantages of the cocyclic approach. The obtained decomposition shows that the determination of Hadamard equivalence classes can be reduced from the full space of $δ$-compatible pairs to the smaller space of ordinary $2$-cocycles, eliminating a redundant factor of $2^n$ from the search space. Computational results for all groups of orders $4$ and $8$, together with a non-associative loop of order $8$, show that the proposed three-dimensional construction refines the classical cocyclic Hadamard classification. While all examples collapse into a single equivalence class in dimension two, they split into several distinct classes in dimension three. In particular, the non-associative loop produces a three-dimensional Hadamard class that does not arise from any of the groups considered, showing that the multidimensional construction detects structural information that is invisible at the cocyclic matrix level.
