How many multiplications do small matrix products need?

Find schemes that multiply small matrices with fewer multiplications, such as 3x3 with fewer than 23 or 5x5 with fewer than 93, or prove lower bounds.

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State the problem, from Yet another catalogue of fast matrix multiplication algorithms

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	1	+# How many multiplications do small matrix products need?
	2	+
	3	+Strassen showed in 1969 that two $2 \times 2$ matrices can be multiplied with 7 multiplications instead of 8, and every such saving for a small format becomes a faster algorithm for large matrices. For $3 \times 3$ matrices the answer is still unknown, somewhere between 19 and 23. Sedoglavic's catalogue records the best known schemes for all formats up to $32 \times 32$, and a new scheme is verified exactly in seconds.
	4	+
	5	+## Problem
	6	+
	7	+Multiplying an $n \times m$ matrix by an $m \times p$ matrix is a bilinear map, described by the matrix multiplication tensor $\langle n, m, p \rangle$. Its rank $R(\langle n, m, p \rangle)$ is the smallest $r$ such that the product can be computed with $r$ multiplications, each of a linear form in the entries of the first matrix by a linear form in the entries of the second, followed by linear combinations; equivalently, the tensor is a sum of $r$ rank-one tensors. Commutativity of the entries may not be used, so the scheme works for matrices of blocks and can be applied recursively.
	8	+
	9	+Task: find schemes with fewer multiplications than the catalogue's entries, or prove better lower bounds, for small formats such as $\langle 3,3,3 \rangle$, $\langle 4,4,4 \rangle$ and $\langle 5,5,5 \rangle$. The coefficient ring matters (integers, rationals, complex numbers or $\mathbb{F}_2$): some records hold only over particular rings.
	10	+
	11	+## Background
	12	+
	13	+A scheme $\langle n, n, n : r \rangle$ applied recursively multiplies $N \times N$ matrices in $O(N^{\log_n r})$ operations, so small-format records feed directly into practical fast algorithms, while the asymptotic exponent of matrix multiplication is studied by other methods. Strassen's 7 multiplications for $2 \times 2$ are optimal (Winograd, 1971). The catalogue gathers 5426 schemes, with explicit coefficients, references and symmetry groups. Since 2022 many entries have been improved by computer search, including reinforcement learning, flip-graph random walks and evolutionary program search.
	14	+
	15	+## What is known
	16	+
	17	+From the catalogue (October 2026):
	18	+
	19	+- $\langle 2,2,2 \rangle$: 7 (Strassen, 1969), optimal.
	20	+- $\langle 3,3,3 \rangle$: 23 (Laderman, 1976). The best general lower bound is 19 (Bläser, 2003); over $\mathbb{F}_2$ it was raised to 20 (Wang, March 2026) and then to 21 in two September 2026 preprints, not yet refereed (Wang; Yang et al., whose proof is formalized in Lean).
	21	+- $\langle 4,4,4 \rangle$: 49 from applying Strassen's scheme twice; 47 over $\mathbb{F}_2$ (Fawzi et al., 2022); 48 with complex coefficients (Novikov et al., 2025, the catalogue's entry) and then with rational coefficients (Dumas, Pernet and Sedoglavic, 2025).
	22	+- Other formats: $\langle 3,4,5 \rangle$: 47 and $\langle 4,5,5 \rangle$: 76 (Fawzi et al., 2022); $\langle 4,4,5 \rangle$: 61 (Novikov et al., 2025); $\langle 5,5,5 \rangle$: 93 and $\langle 6,6,6 \rangle$: 153 (Moosbauer and Poole, 2025); $\langle 5,5,6 \rangle$: 110 (Kauers and Wood, 2025).
	23	+
	24	+## What would count as progress
	25	+
	26	+- A scheme with fewer multiplications for some format and coefficient ring, given as the three factor matrices of a tensor decomposition. Expanding the decomposition and comparing it with the matrix multiplication tensor is an exact computation that takes seconds even for the largest catalogue formats.
	27	+- A better lower bound on a small rank, such as $R(\langle 3,3,3 \rangle) \ge 20$ over the rationals, with a proof or a machine-checkable certificate.
	28	+- Schemes that match a record with fewer additions or better numerical stability, which the catalogue also tracks for practical use.
	29	+
	30	+## References
	31	+
	32	+1. V. Strassen, "Gaussian elimination is not optimal", Numerische Mathematik 13 (1969), 354–356. [doi:10.1007/BF02165411](https://doi.org/10.1007/BF02165411)
	33	+2. J. D. Laderman, "A noncommutative algorithm for multiplying 3×3 matrices using 23 multiplications", Bulletin of the AMS 82 (1976), 126–128. [doi:10.1090/S0002-9904-1976-13988-2](https://doi.org/10.1090/S0002-9904-1976-13988-2)
	34	+3. M. Bläser, "On the complexity of the multiplication of matrices of small formats", Journal of Complexity 19 (2003), 43–60. [doi:10.1016/s0885-064x(02)00007-9](https://doi.org/10.1016/s0885-064x(02)00007-9)
	35	+4. A. Fawzi et al., "Discovering faster matrix multiplication algorithms with reinforcement learning", Nature 610 (2022), 47–53. [doi:10.1038/s41586-022-05172-4](https://doi.org/10.1038/s41586-022-05172-4)
	36	+5. J. Moosbauer and M. Poole, "Flip graphs with symmetry and new matrix multiplication schemes" (2025). [arXiv:2502.04514](https://arxiv.org/abs/2502.04514)
	37	+6. C. Wang, "Automated lower bounds for bilinear complexity over finite fields" (2026). [arXiv:2603.07280](https://arxiv.org/abs/2603.07280)
	38	+
	39	+## Source
	40	+
	41	+Imported from [Yet another catalogue of fast matrix multiplication algorithms](https://fmm.univ-lille.fr/), curated by Alexandre Sedoglavic. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.