New fast matrix multiplication schemes

Bilinear schemes for multiplying an n1×n2 matrix by an n2×n3 matrix, all with coefficients in {-1, 0, 1} (ZT), which makes them valid over any ring.

Previous values are the lower of the FastMatrixMultiplication table as of 2026-09-29 and the Sedoglavic catalogue as of 2026-09-30. Values that the table took over from earlier releases of this dataset are not counted as previous values.

Below the best known rank in any ring

Format	Rank	Previous best (any ring)	Previous ZT
7×7×9	314	315	316
6×11×13	572	574	584
8×10×13	684	686	686
8×10×14	724	726	726
7×11×15	769	777	778
9×11×13	833	835	843
10×13×14	1150	1152	1152
11×13×15	1362	1371	1377
11×14×14	1367	1376	1376
13×13×15	1591	1605	1605

New ZT records

Format	Rank	Previous ZT	Best known (any ring)
2×12×15	280	281	278
2×13×15	304	305	300
2×13×16	324	325	320
2×15×16	374	375	368
4×11×15	457	458	449
6×11×11	492	496	490
7×9×14	598	600	597
6×11×14	617	621	613
7×9×15	638	639	634
8×11×16	906	914	904
9×14×16	1260	1270	1254
9×15×15	1269	1276	1236
8×16×16	1256	1260	1230
12×12×15	1326	1332	1280
10×14×16	1416	1418	1398
12×13×15	1464	1470	1442
11×15×15	1547	1548	1540
11×15×16	1641	1657	1605
13×13×16	1709	1711	1704
11×16×16	1749	1752	1724
13×14×16	1805	1820	1796

Files

schemes/ holds one file per scheme in the JSON format of the FastMatrixMultiplication repository: n, m, u, v, w and the readable multiplications and elements; complexity is the naive addition count. SHA256SUMS lists the checksum of every file.

Verification

Every scheme satisfies the Brent equations exactly:

python verify.py schemes/*.json

verify.py uses exact rational arithmetic and nothing but the Python standard library. The schemes were also accepted by validate_schemes from ternary_flip_graph.

License and citation

The schemes are licensed under CC BY 4.0. Please cite as described in CITATION.cff.