A searchable catalogue that assembles 5,426 upper bounds for fast (non‑commutative) matrix multiplication across nearly all matrix formats from 2×2 up to 32×32. Each entry is given in the ⟨X×Y×Z:N⟩ notation meaning an X×Y times Y×Z product requiring N coefficient multiplications; table columns report the tensor decomposition rank (N), the naive rank, how the algorithm is constructed (explicit formula, sum, tensor/Kronecker products, or references), a ratio measuring efficiency, an asymptotic complexity/exponent value, and any known stabilizer (symmetry) group. Entries link to explicit decompositions or bibliographic sources and a JSON summary of tensor ranks is available in a companion repository.
Substantively, the collection aggregates classical results (e.g., Strassen 1969, Hopcroft-Kerr 1971) together with modern discoveries (including machine‑discovered and recent papers) and many composed constructions (direct sums, Kronecker/tensor products, and automated reconstructions). The dataset was reconstructed computationally and documents exact constructions, ranks versus naive bounds, and symmetry information that matters for optimization and theoretical bounds on the matrix multiplication exponent. The catalogue is intended as a practical reference for researchers seeking explicit fast algorithms, upper bounds by format, and pointers to constructions and provenance.
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