A computational framework for multidimensional loop-cocyclic Hadamard matrices

Source abstract

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.