FastMatrixMultiplication

A research project investigating fast matrix multiplication algorithms for small matrix formats, from 2×2×2 to 16×16×16. The primary goal is to discover efficient schemes with coefficients restricted to the ternary set {-1, 0, 1}, focusing on all tensor shapes satisfying max(n₁, n₂, n₃) ≤ 16.

Overview

This repository documents the search for fast matrix multiplication (FMM) schemes using a custom meta flip graph method. The search focuses on schemes that use only the coefficients -1, 0, and 1, denoted as ZT. This constraint is significant for practical implementations where computational complexity and hardware efficiency are critical.

Key insight: several known optimal schemes originally found over the rationals (Q) or integers (Z) have been successfully rediscovered with minimal, ternary coefficients. This can lead to more efficient and hardware-friendly implementations.

Latest progress

For a detailed history of discoveries and improvements, see the CHANGELOG.md.

Publications

Key results

New best ranks

New schemes have been discovered that improve the state-of-the-art for matrix multiplication achieving lower ranks than previously known.

Format	Prev rank	New rank	ω
`2×4×11`	71 (`Q`)	70 (`ZT`)	2.846666725
`3×5×9`	104 (`Z`)	102 (`ZT`)	2.828571093
`3×5×10`	115 (`Z`)	114 (`ZT`)	2.835687395
`3×7×9`	142 (`Q`)	141 (`ZT`)	2.832315186
`3×7×13`	205 (`Q`)	204 (`ZT/Q`)	2.844182227
`3×7×14`	220 (`Q`)	219 (`ZT`)	2.844547952
`3×7×15`	236 (`Q`)	235 (`Q`)	2.847205515
`3×9×11`	224 (`Q`)	222 (`Q`)	2.846644652
`3×9×13`	262 (`Q`)	261 (`Q`)	2.848348427
`3×9×14`	283 (`Q`)	281 (`ZT`)	2.850103717
`3×10×11`	249 (`Q`)	248 (`Q`)	2.852219687
`3×10×14`	314 (`Q`)	312 (`ZT`)	2.852364741
`3×10×15`	336 (`Q`)	335 (`ZT`)	2.855080165
`3×10×16`	360 (`Q`)	355 (`ZT`)	2.853411678
`3×11×14`	346 (`Q`)	345 (`Q`)	2.857215849
`3×13×13`	379 (`Q`)	378 (`Q`)	2.858577688
`3×13×14`	408 (`Q`)	407 (`Q`)	2.860150618
`3×13×15`	436 (`Q`)	435 (`Q`)	2.860506655
`3×13×16`	465 (`Q`)	464 (`Q`)	2.861905425
`3×14×14`	440 (`Q`)	438 (`ZT/Q`)	2.861445516
`3×14×15`	470 (`Q`)	469 (`Q`)	2.862645108
`3×14×16`	502 (`Q`)	500 (`ZT/Q`)	2.863761055
`3×15×16`	536 (`Q`)	534 (`ZT`)	2.863728243
`3×16×16`	574 (`Q`)	569 (`Q`)	2.864576103
`4×4×10`	120 (`Q`)	115 (`ZT`)	2.804789925
`4×4×11`	130 (`Q`)	129 (`ZT`)	2.819743225
`4×4×12`	142 (`Q`)	141 (`ZT`)	2.823831239
`4×4×14`	165 (`Q`)	163 (`Q`)	2.823771262
`4×4×15`	177 (`Q`)	176 (`ZT`)	2.830226950
`4×4×16`	189 (`Q`)	188 (`ZT`)	2.832970819
`4×5×9`	136 (`Q`)	132 (`ZT`)	2.820821776
`4×5×10`	151 (`Z`)	146 (`ZT`)	2.821805270
`4×5×11`	165 (`Z`)	160 (`ZT`)	2.822872235
`4×5×12`	180 (`Z`)	174 (`ZT`)	2.823971094
`4×5×13`	194 (`Z`)	191 (`ZT`)	2.833613095
`4×5×14`	208 (`Z`)	206 (`Q`)	2.836597217
`4×5×15`	226 (`Z`)	221 (`ZT`)	2.839254157
`4×5×16`	240 (`Q`)	235 (`ZT`)	2.839432229
`4×6×13`	228 (`Z`)	227 (`ZT`)	2.833857047
`4×6×16`	280 (`ZT`)	276 (`ZT`)	2.833509566
`4×7×8`	164 (`ZT`)	161 (`ZT`)	2.816927225
`4×7×11`	227 (`Z`)	224 (`Q`)	2.833273201
`4×7×12`	246 (`Z`)	242 (`ZT`)	2.830754429
`4×7×13`	266 (`Q`)	265 (`ZT`)	2.838520048
`4×7×14`	285 (`ZT`)	284 (`ZT`)	2.838080655
`4×7×15`	307 (`Q`)	305 (`ZT/Q`)	2.841094648
`4×7×16`	324 (`ZT`)	322 (`ZT`)	2.837713576
`4×8×8`	182 (`ZT`)	180 (`ZT`)	2.809444911
`4×9×10`	255 (`ZT`)	250 (`ZT`)	2.814150526
`4×9×11`	280 (`ZT`)	275 (`ZT`)	2.817111923
`4×9×13`	329 (`Q`)	325 (`ZT`)	2.822080998
`4×9×14`	355 (`Z`)	350 (`ZT/Z`)	2.824199930
`4×9×16`	400 (`ZT`)	398 (`ZT`)	2.825527347
`4×10×15`	417 (`Q`)	413 (`Q`)	2.824846255
`4×11×12`	365 (`Z`)	362 (`ZT/Q`)	2.819374888
`4×11×15`	452 (`Q`)	449 (`Q`)	2.821995048
`4×11×16`	489 (`Q`)	480 (`ZT`)	2.824765046
`4×12×12`	390 (`ZT`)	389 (`ZT`)	2.814731749
`4×12×13`	426 (`Q`)	422 (`Q`)	2.817680586
`4×12×14`	456 (`Q`)	452 (`Q`)	2.817253261
`4×12×16`	520 (`ZT`)	513 (`ZT`)	2.817793502
`4×13×15`	528 (`Q`)	521 (`Q`)	2.818204205
`4×13×16`	568 (`ZT`)	560 (`Z`)	2.823361605
`4×14×15`	568 (`Q`)	557 (`Q`)	2.816955831
`4×14×16`	610 (`ZT`)	598 (`ZT`)	2.821556397
`4×15×15`	600 (`ZT`)	596 (`Q`)	2.818231324
`4×15×16`	640 (`Q`)	632 (`Q`)	2.817366557
`4×16×16`	676 (`ZT`)	666 (`ZT`)	2.813813510
`5×5×9`	167 (`Z`)	161 (`ZT`)	2.814610506
`5×5×10`	184 (`Q`)	178 (`ZT`)	2.815441580
`5×5×11`	202 (`Q`)	195 (`ZT`)	2.816386568
`5×5×12`	220 (`Z`)	204 (`ZT`)	2.797154354
`5×5×13`	237 (`Z`)	227 (`ZT`)	2.813855803
`5×5×14`	254 (`Z`)	244 (`ZT`)	2.815242892
`5×5×15`	271 (`Q`)	262 (`ZT`)	2.818498736
`5×5×16`	288 (`Q`)	280 (`ZT`)	2.821408468
`5×6×9`	197 (`Z`)	193 (`ZT`)	2.820092998
`5×6×10`	218 (`Z`)	216 (`ZT`)	2.827217780
`5×7×8`	205 (`Q`)	204 (`ZT`)	2.831402964
`5×9×9`	294 (`Q`)	293 (`ZT`)	2.838247561
`5×9×15`	474 (`Z`)	468 (`ZT`)	2.831345872
`5×10×12`	413 (`Z`)	408 (`ZT`)	2.819133943
`5×11×12`	455 (`Q`)	454 (`Q`)	2.827112377
`5×12×15`	615 (`Q`)	612 (`ZT`)	2.829914687
`5×12×16`	656 (`Q`)	655 (`Q`)	2.832983066
`5×13×13`	588 (`Q`)	587 (`Q`)	2.837827448
`5×13×14`	630 (`Q`)	628 (`Q`)	2.836688582
`5×14×14`	676 (`Q`)	672 (`ZT/Z`)	2.835662569
`5×15×15`	762 (`ZT`)	761 (`ZT`)	2.833078290
`6×6×13`	316 (`Q`)	315 (`Q`)	2.806832057
`6×7×7`	215 (`ZT`)	212 (`ZT`)	2.827400948
`6×7×8`	239 (`ZT`)	238 (`ZT`)	2.822158898
`6×7×9`	270 (`ZT`)	264 (`ZT`)	2.818558639
`6×7×10`	296 (`Z`)	293 (`Q`)	2.821158816
`6×8×16`	511 (`Q`)	510 (`ZT`)	2.815145110
`6×9×9`	342 (`Z`)	332 (`Q`)	2.815198446
`6×9×10`	373 (`Z`)	367 (`Q`)	2.815845324
`6×9×11`	407 (`Q`)	404 (`Q`)	2.818942356
`6×9×12`	434 (`Q`)	429 (`Q`)	2.808878248
`6×9×13`	474 (`Q`)	468 (`Q`)	2.814402245
`6×9×14`	500 (`Q`)	494 (`Q`)	2.807406517
`6×9×15`	532 (`Q`)	529 (`Q`)	2.809148737
`6×9×16`	556 (`Q`)	552 (`Q`)	2.801218708
`6×10×15`	597 (`Q`)	594 (`ZT`)	2.816748899
`6×11×12`	524 (`Q`)	521 (`ZT`)	2.811757811
`6×11×15`	661 (`Z`)	653 (`ZT`)	2.819014665
`6×11×16`	695 (`Q`)	684 (`Q`)	2.812868273
`6×12×14`	658 (`Q`)	645 (`Q`)	2.806322591
`6×12×15`	705 (`Z`)	686 (`Q`)	2.805072101
`6×12×16`	746 (`Q`)	736 (`ZT/Z`)	2.809331029
`6×13×13`	680 (`Q`)	678 (`Q`)	2.825542860
`6×13×14`	730 (`Q`)	726 (`Q`)	2.824944345
`6×13×15`	771 (`Z`)	763 (`ZT/Z`)	2.818464723
`6×13×16`	819 (`Q`)	816 (`Z`)	2.821209653
`6×14×14`	777 (`Q`)	776 (`Q`)	2.823594480
`6×14×15`	825 (`Q`)	814 (`ZT/Z`)	2.816396649
`6×14×16`	880 (`Q`)	864 (`ZT`)	2.815990055
`6×15×15`	870 (`ZT`)	859 (`Q`)	2.811834182
`6×15×16`	928 (`Q`)	920 (`ZT/Z`)	2.815181444
`6×16×16`	988 (`ZT`)	972 (`ZT`)	2.812899669
`7×7×10`	346 (`Z`)	345 (`Q`)	2.830075228
`7×8×9`	350 (`Q`)	347 (`ZT/Z`)	2.820049700
`7×8×12`	454 (`Q`)	452 (`ZT/Z`)	2.817253261
`7×8×15`	571 (`Q`)	557 (`Q`)	2.816955831
`7×8×16`	603 (`Q`)	598 (`ZT`)	2.821556397
`7×9×9`	398 (`Q`)	396 (`ZT`)	2.830161790
`7×9×10`	437 (`Z`)	433 (`ZT/Z`)	2.825473910
`7×9×11`	480 (`Q`)	478 (`ZT`)	2.829651018
`7×9×12`	510 (`Q`)	508 (`Q`)	2.820055471
`7×9×15`	639 (`Z`)	634 (`Q`)	2.825226157
`7×10×12`	564 (`Z`)	557 (`Q`)	2.816955831
`7×10×15`	711 (`Q`)	694 (`ZT/Z`)	2.821431419
`7×10×16`	752 (`Q`)	736 (`Q`)	2.820602981
`7×11×15`	778 (`Z`)	777 (`Q`)	2.831357038
`7×11×16`	827 (`Q`)	822 (`ZT/Z`)	2.829413433
`7×12×15`	831 (`Z`)	815 (`ZT/Z`)	2.816912591
`7×12×16`	880 (`Q`)	878 (`Z`)	2.822684315
`7×13×13`	795 (`Q`)	794 (`Q`)	2.830948485
`7×13×14`	852 (`Q`)	850 (`Q`)	2.830202017
`7×13×16`	968 (`Q`)	962 (`Q`)	2.829297704
`7×14×14`	912 (`Q`)	909 (`ZT/Z`)	2.829037251
`7×14×15`	976 (`Z`)	952 (`ZT/Z`)	2.821286881
`7×14×16`	1034 (`Q`)	1022 (`Q`)	2.825469455
`7×15×16`	1099 (`Q`)	1083 (`Q`)	2.822639497
`8×8×15`	635 (`Q`)	628 (`Q`)	2.814592730
`8×8×16`	672 (`Q`)	666 (`ZT`)	2.813813510
`8×9×10`	487 (`ZT`)	482 (`ZT/Z`)	2.817012414
`8×9×11`	533 (`Q`)	521 (`ZT`)	2.811757811
`8×9×13`	624 (`Z`)	617 (`ZT/Z`)	2.817259628
`8×9×14`	669 (`Z`)	654 (`ZT/Z`)	2.812333691
`8×9×15`	705 (`Z`)	699 (`ZT/Z`)	2.813135327
`8×9×16`	746 (`Q`)	735 (`ZT`)	2.808752406
`8×10×12`	630 (`Z`)	624 (`Q`)	2.811801179
`8×10×15`	789 (`Z`)	778 (`Q`)	2.816637962
`8×10×16`	832 (`Q`)	822 (`Q`)	2.814298209
`8×11×12`	680 (`Q`)	676 (`Q`)	2.807798855
`8×11×15`	859 (`Z`)	848 (`Q`)	2.815247371
`8×11×16`	920 (`Q`)	904 (`Q`)	2.816647975
`8×12×13`	798 (`Q`)	784 (`Q`)	2.804375455
`8×12×14`	861 (`Z`)	843 (`Q`)	2.805742480
`8×12×15`	915 (`ZT`)	904 (`Q`)	2.807944089
`8×13×15`	1005 (`ZT`)	992 (`ZT/Q`)	2.815278496
`8×13×16`	1064 (`Q`)	1054 (`Q`)	2.815302758
`8×14×14`	1008 (`Z`)	1004 (`Q`)	2.818224059
`8×14×15`	1080 (`ZT`)	1063 (`ZT/Q`)	2.815109808
`8×14×16`	1138 (`Q`)	1104 (`Q`)	2.806012534
`8×15×15`	1140 (`ZT`)	1130 (`Q`)	2.813661626
`8×15×16`	1198 (`Q`)	1185 (`Q`)	2.808501081
`8×16×16`	1248 (`Q`)	1230 (`Q`)	2.799393436
`9×9×9`	498 (`ZT`)	486 (`ZT`)	2.815464877
`9×9×14`	726 (`Q`)	720 (`Q`)	2.806246597
`9×9×15`	783 (`Q`)	760 (`Q`)	2.801824302
`9×9×16`	825 (`Q`)	823 (`Q`)	2.809929143
`9×10×10`	600 (`Z`)	597 (`ZT`)	2.818970672
`9×10×12`	684 (`Q`)	668 (`Q`)	2.793651686
`9×10×13`	772 (`Z`)	758 (`Q`)	2.815672846
`9×10×14`	820 (`Z`)	808 (`Q`)	2.813287623
`9×10×15`	870 (`ZT`)	865 (`Z`)	2.814731263
`9×10×16`	939 (`Q`)	916 (`Q`)	2.813383975
`9×11×11`	725 (`Q`)	715 (`Q`)	2.819505933
`9×11×12`	760 (`Q`)	738 (`Q`)	2.798270808
`9×11×13`	849 (`Z`)	835 (`Q`)	2.818729064
`9×11×14`	904 (`Z`)	882 (`Q`)	2.812562599
`9×11×15`	981 (`Q`)	958 (`ZT`)	2.819945748
`9×11×16`	1030 (`Z`)	996 (`Z`)	2.811083198
`9×12×13`	900 (`Q`)	878 (`Q`)	2.805673201
`9×12×14`	945 (`Q`)	940 (`Q`)	2.805232985
`9×12×15`	1000 (`Q`)	996 (`Q`)	2.802534955
`9×12×16`	1080 (`Q`)	1035 (`Q`)	2.793729377
`9×13×13`	996 (`Z`)	981 (`Q`)	2.820440786
`9×13×14`	1063 (`Z`)	1026 (`Q`)	2.810379562
`9×13×15`	1135 (`Q`)	1119 (`ZT/Z`)	2.819269106
`9×13×16`	1210 (`Z`)	1179 (`Q`)	2.815916926
`9×14×14`	1136 (`Z`)	1101 (`Q`)	2.810831956
`9×14×15`	1185 (`Q`)	1175 (`Q`)	2.810993734
`9×14×16`	1260 (`Q`)	1254 (`Q`)	2.812806553
`9×15×15`	1290 (`Q`)	1236 (`Q`)	2.805463706
`9×15×16`	1350 (`Z`)	1320 (`ZT/Z`)	2.807572842
`9×16×16`	1444 (`ZT`)	1380 (`ZT/Z`)	2.801393711
`10×10×12`	770 (`Z`)	768 (`ZT`)	2.811164062
`10×11×12`	850 (`Z`)	849 (`Q`)	2.815739433
`10×11×15`	1067 (`Q`)	1050 (`ZT/Z`)	2.816973777
`10×11×16`	1136 (`Q`)	1112 (`ZT`)	2.815676689
`10×12×12`	910 (`Z`)	902 (`ZT/Z`)	2.807030426
`10×12×15`	1140 (`ZT`)	1122 (`ZT/Z`)	2.810818006
`10×12×16`	1216 (`Q`)	1176 (`Q`)	2.805475746
`10×13×15`	1242 (`Z`)	1230 (`ZT/Z`)	2.817513014
`10×13×16`	1332 (`Z`)	1318 (`Q`)	2.820846164
`10×14×15`	1327 (`Z`)	1314 (`Z`)	2.816125387
`10×14×16`	1423 (`Z`)	1398 (`Q`)	2.816663561
`10×15×15`	1395 (`Z`)	1385 (`Q`)	2.811406977
`10×15×16`	1497 (`Z`)	1482 (`Q`)	2.814186431
`10×16×16`	1586 (`ZT`)	1560 (`Q`)	2.810651214
`11×11×12`	936 (`Z`)	922 (`Q`)	2.812867384
`11×11×15`	1181 (`Z`)	1169 (`ZT`)	2.824115356
`11×11×16`	1236 (`Q`)	1230 (`ZT`)	2.820195393
`11×12×12`	990 (`Q`)	968 (`Q`)	2.799472327
`11×12×13`	1102 (`Z`)	1082 (`Q`)	2.814231919
`11×12×14`	1182 (`Z`)	1153 (`Q`)	2.811853672
`11×12×15`	1264 (`Q`)	1234 (`Z`)	2.813129312
`11×12×16`	1312 (`Q`)	1278 (`Q`)	2.803143027
`11×13×13`	1210 (`Z`)	1205 (`ZT`)	2.827216655
`11×13×14`	1298 (`Z`)	1292 (`ZT`)	2.827166171
`11×13×15`	1377 (`Z`)	1371 (`Q`)	2.824949046
`11×13×16`	1472 (`Z`)	1446 (`Q`)	2.822035717
`11×14×14`	1388 (`Z`)	1376 (`ZT`)	2.824489318
`11×14×15`	1471 (`Z`)	1432 (`ZT/Z`)	2.814780394
`11×14×16`	1571 (`Q`)	1520 (`Q`)	2.814428649
`11×15×16`	1656 (`Q`)	1605 (`Q`)	2.810502126
`12×12×13`	1152 (`Q`)	1144 (`Q`)	2.803918249
`12×12×14`	1250 (`Q`)	1234 (`Q`)	2.806467563
`12×12×16`	1392 (`Q`)	1380 (`ZT/Z`)	2.801393711
`12×13×14`	1382 (`Q`)	1370 (`Q`)	2.818044255
`12×13×15`	1460 (`Q`)	1442 (`Q`)	2.812789736
`12×13×16`	1556 (`Q`)	1544 (`Z`)	2.815794289
`12×14×14`	1481 (`Q`)	1449 (`Q`)	2.812807792
`12×14×15`	1540 (`Q`)	1538 (`Q`)	2.810862435
`12×14×16`	1638 (`Q`)	1617 (`Q`)	2.806918970
`12×15×16`	1728 (`Q`)	1725 (`ZT/Z`)	2.806957387
`12×16×16`	1824 (`Q`)	1815 (`Q`)	2.803398069
`13×13×13`	1426 (`Q`)	1421 (`Q`)	2.830120644
`13×13×14`	1524 (`Z`)	1511 (`ZT`)	2.826838093
`13×13×16`	1713 (`Q`)	1704 (`Q`)	2.824705676
`13×14×14`	1625 (`Z`)	1614 (`ZT`)	2.825351482
`13×14×15`	1714 (`Z`)	1681 (`ZT/Z`)	2.816136526
`13×14×16`	1825 (`Q`)	1800 (`Q`)	2.819075643
`13×15×15`	1803 (`Z`)	1797 (`Z`)	2.816875265
`13×15×16`	1932 (`Z`)	1908 (`ZT/Q`)	2.816628414
`14×14×15`	1813 (`Z`)	1798 (`ZT/Z`)	2.815280055
`14×14×16`	1939 (`Q`)	1931 (`Q`)	2.819303950
`14×15×15`	1905 (`Z`)	1890 (`Q`)	2.809752096
`14×16×16`	2142 (`Q`)	2128 (`Q`)	2.808914234
`15×15×16`	2173 (`Q`)	2132 (`ZT/Z`)	2.808074285

Rediscovery in the ternary coefficient set (`ZT`)

The following schemes have been rediscovered in the ZT format. Originally known over the rational (Q) or integer (Z) fields, implementations with coefficients restricted to the ternary set were previously unknown.

Format	Rank	Known ring
`2×3×10`	50	`Z`
`2×3×13`	65	`Z`
`2×3×15`	75	`Z`
`2×4×6`	39	`Z`
`2×4×9`	58	`Q`
`2×4×10`	64	`Q`
`2×4×12`	77	`Q`
`2×4×13`	83	`Q`
`2×4×15`	96	`Q`
`2×5×7`	55	`Q`
`2×5×8`	63	`Q`
`2×5×9`	72	`Q`
`2×5×10`	79	`Q`
`2×5×13`	102	`Q`
`2×5×14`	110	`Q`
`2×5×15`	118	`Q`
`2×5×16`	126	`Q`
`2×6×6`	56	`Z`
`2×6×7`	66	`Z/Q`
`2×6×8`	75	`Q`
`2×6×9`	86	`Z`
`2×6×11`	103	`Z`
`2×6×12`	112	`Z`
`2×6×13`	122	`Q`
`2×6×14`	131	`Q`
`2×6×16`	150	`Z`
`2×7×7`	76	`Q`
`2×7×8`	88	`Z`
`2×7×10`	110	`Z`
`2×7×11`	121	`Z`
`2×7×12`	131	`Q`
`2×7×13`	142	`Q`
`2×7×14`	152	`Q`
`2×7×15`	164	`Q`
`2×8×9`	113	`Q`
`2×8×11`	138	`Z`
`2×8×12`	150	`Z`
`2×8×14`	175	`Q`
`2×8×15`	188	`Z`
`3×3×7`	49	`Q`
`3×3×9`	63	`Q`
`3×3×10`	69	`Q`
`3×3×11`	76	`Q`
`3×4×5`	47	`Z`
`3×4×6`	54	`Z/Q`
`3×4×8`	73	`Q`
`3×4×9`	83	`Q`
`3×4×10`	92	`Q`
`3×4×11`	101	`Q`
`3×4×12`	108	`Q`
`3×4×16`	146	`Q`
`3×5×6`	68	`Z`
`3×5×7`	79	`Q`
`3×5×8`	90	`Z/Q`
`3×5×11`	126	`Z`
`3×5×12`	136	`Z`
`3×5×13`	147	`Q`
`3×5×14`	158	`Q`
`3×5×15`	169	`Q`
`3×5×16`	180	`Z`
`3×6×8`	108	`Z/Q`
`3×7×7`	111	`Q`
`3×8×9`	163	`Q`
`3×8×10`	180	`Z`
`3×8×11`	198	`Q`
`3×8×12`	216	`Q`
`3×8×15`	270	`Z`
`3×8×16`	288	`Q`
`3×11×11`	274	`Q`
`4×4×6`	73	`Z/Q`
`4×4×8`	96	`Q`
`4×5×6`	90	`Z`
`4×5×7`	104	`Z/Q`
`4×5×8`	118	`Z/Q`
`4×6×7`	123	`Z/Q`
`4×6×9`	159	`Q`
`4×6×10`	175	`Z`
`4×6×11`	194	`Q`
`4×6×15`	263	`Z`
`4×7×7`	144	`Z/Q`
`4×8×13`	297	`Z`
`4×9×15`	375	`Z`
`4×10×13`	361	`Q`
`4×10×14`	385	`Q`
`4×10×16`	441	`Q`
`4×11×11`	340	`Z`
`4×11×14`	429	`Q`
`4×14×14`	532	`Q`
`5×5×6`	110	`Z/Q`
`5×5×7`	127	`Z/Q`
`5×5×8`	144	`Z/Q`
`5×6×6`	130	`Z/Q`
`5×6×7`	150	`Z/Q`
`5×6×8`	170	`Z/Q`
`5×6×16`	340	`Q`
`5×7×7`	176	`Z/Q`
`5×7×9`	229	`Q`
`5×7×10`	254	`Z`
`5×7×11`	277	`Z`
`5×7×13`	325	`Q`
`5×8×9`	260	`Q`
`5×8×12`	333	`Q`
`5×8×16`	445	`Q`
`5×9×10`	322	`Q`
`5×9×11`	353	`Q`
`5×9×12`	377	`Q`
`5×10×11`	386	`Z`
`5×10×13`	451	`Q`
`5×10×14`	481	`Q`
`5×10×15`	519	`Q`
`5×10×16`	549	`Q`
`5×11×16`	609	`Q`
`5×15×16`	813	`Z`
`6×6×7`	183	`Z/Q`
`6×8×10`	329	`Z`
`6×8×11`	357	`Q`
`6×8×12`	378	`Q`
`6×10×11`	446	`Z`
`6×10×12`	476	`Z`
`6×10×13`	520	`Q`
`6×10×14`	553	`Q`
`6×10×16`	630	`Q`
`7×8×10`	385	`Z`
`7×8×11`	423	`Q`
`7×10×11`	526	`Z`
`7×10×13`	614	`Q`
`7×10×14`	653	`Q`
`7×13×15`	909	`Z`
`8×8×11`	475	`Q`
`8×8×13`	559	`Q`
`8×10×11`	588	`Z`
`8×10×13`	686	`Z`
`8×10×14`	728	`Z`
`8×11×14`	804	`Z`
`8×12×16`	960	`Q`
`8×13×14`	945	`Z`
`10×10×10`	651	`Z`
`10×10×11`	719	`Z`
`10×10×13`	838	`Z`
`10×10×14`	889	`Z`
`10×10×15`	957	`Q`
`10×10×16`	1008	`Q`
`10×11×11`	793	`Z`
`10×11×13`	924	`Z`
`10×11×14`	981	`Z`
`10×12×13`	990	`Z`
`10×12×14`	1050	`Z`
`10×13×13`	1082	`Z`
`10×13×14`	1154	`Z`
`10×14×14`	1232	`Z`
`11×11×11`	873	`Z`
`11×11×13`	1023	`Z`
`11×11×14`	1093	`Z`
`13×13×15`	1605	`Z`
`15×15×15`	2058	`Q`

Rediscovery in the integer ring (`Z`)

The following schemes, originally known over the rational field (Q), have now been rediscovered in the integer ring (Z). Implementations restricted to integer coefficients were previously unknown.

Format	Rank
`2×7×9`	99
`2×7×16`	175
`2×11×12`	204
`2×11×13`	221
`2×11×14`	238
`2×13×15`	300
`2×13×16`	320
`2×15×16`	368

Methodology & instruments

The research employs a multi-stage approach using custom-built tools:

ternary_flip_graph: core flip graph exploration toolkit

A comprehensive CPU-based toolkit for discovering fast matrix multiplication algorithms using flip graph techniques. Supports multiple coefficient sets ({0, 1}, {0, 1, 2}, {-1, 0, 1}) and provides tools for rank minimization, complexity optimization, alternative scheme discovery, and meta operations for transforming schemes between dimensions.

ternary_addition_reducer: addition reduction tool

A high-performance tool for optimizing the number of arithmetic additions in fast matrix multiplication algorithms with ternary coefficients. It implements multiple heuristic strategies to find near-optimal computation schemes, significantly reducing the additive cost of matrix multiplications schemes.

Alternative scheme finding

This script starts from an existing binary (Z2) scheme and discovers new, non-identical schemes for the same dimensions. It works by:

Randomly preserving coefficients from the original U, V, W matrices with configurable probabilities;
Solving the resulting Brent equations using the CryptoMiniSat SAT solver;
Exploring the solution space around known schemes.

python find_alternative_schemes.py -i <input_scheme_path> -o <output_dir> [options]

Options:

-pu, -pv, -pw - probability thresholds for preserving U, V, W coefficients (default: 0.8)
--max-time - sat solver timeout in seconds (default: 20)
-f - maximum flip iterations for more effective search
-t - number of sat solver threads

Ternary coefficient Lifting

This script lifts binary (Z2) schemes to the ternary integer coefficient set (ZT, coefficients {-1, 0, 1}) using OR-Tools SAT solver.

python lift_schemes.py -i <input_dir> -o <output_dir> [options]

Options:

--max-time - maximum lifting time per scheme in seconds
--max-solutions - maximum number of ternary solutions to find
--sort-scheme - output schemes in "canonical" form
-f - force re-lifting of existing schemes

Analyzed Schemes & Data Sources

This research consolidates and analyzes schemes from several leading sources in the field:

Source	Description
FMM catalogue	The central repository for known fast matrix multiplication algorithms (fmm.univ-lille.fr).
Alpha Tensor	Schemes from DeepMind's AlphaTensor project (https://github.com/google-deepmind/alphatensor/tree/main/algorithms).
Alpha Evolve	Schemes from DeepMind's AlphaEvolve project (mathematical_results.ipynb).
Original Flip Graph	Foundational work by Jakob Moosbauer (flips).
Adaptive flip graph	Improved flip graph approach (adap).
Symmetric flip graph	Flip graphs with symmetry (symmetric-flips).
Meta Flip Graph	Advanced flip graph techniques by M. Kauers et al. (matrix-multiplication).
FMM Add Reduction	Work on additive reductions by @werekorren (fmm_add_reduction).

Scheme File Formats

This repository uses two JSON formats for storing matrix-multiplication schemes:

Full scheme format (.json) - complete description with human-readable bilinear products and the matrices U, V, W;
Reduced scheme format (_reduced.json) - compact representation used after additive-complexity reduction.

Both formats are described below.

Full scheme format

This is the primary format used in the repository. Each file describes a bilinear algorithm for multiplying an n₁×n₂ by n₂×n₃ using mmultiplications.

Top level structure

{
    "n": [n₁, n₂, n₃],
    "m": rank,
    "z2": false,
    "u": [...],
    "v": [...],
    "w": [...],
    "multiplications": [...],
    "elements": [...]
}

Fields

n - array [n₁, n₂, n₃] describing the dimensions (A is n₁ × n₂, B is n₂ × n₃);
m - number of bilinear multiplications (rank);
z2 - whether coefficients are in Z2 field (true) or in any other (false);
multiplications (human-readable) - list of expressions m_k = (linear form in A) * (linear form in B);
elements (human-readable) - expressions for each entry c_{ij} as linear combination of the m_k;
u (machine-readable) - matrix encoding the linear form of A, size m × (n₁·n₂);
v (machine-readable) - matrix encoding the linear form of B, size m × (n₂·n₃);
w (machine-readable) - matrix encoding the linear form of Cᵀ, size m × (n₃·n₁);

This format is intended for reproducibility and human and machine readability.

Example

Scheme 2×2×2:7:

{
    "n": [2, 2, 2],
    "m": 7,
    "z2": false,
    "multiplications": [
        "m1 = (a11 + a22) * (b11 + b22)",
        "m2 = (a12 - a22) * (b21 + b22)",
        "m3 = (-a11 + a21) * (b11 + b12)",
        "m4 = (a11 + a12) * (b22)",
        "m5 = (a11) * (b12 - b22)",
        "m6 = (a22) * (-b11 + b21)",
        "m7 = (a21 + a22) * (b11)"
    ],
    "elements": [
        "c11 = m1 + m2 - m4 + m6",
        "c12 = m4 + m5",
        "c21 = m6 + m7",
        "c22 = m1 + m3 + m5 - m7"
    ],
    "u": [
        [1, 0, 0, 1],
        [0, 1, 0, -1],
        [-1, 0, 1, 0],
        [1, 1, 0, 0],
        [1, 0, 0, 0],
        [0, 0, 0, 1],
        [0, 0, 1, 1]
    ],
    "v": [
        [1, 0, 0, 1],
        [0, 0, 1, 1],
        [1, 1, 0, 0],
        [0, 0, 0, 1],
        [0, 1, 0, -1],
        [-1, 0, 1, 0],
        [1, 0, 0, 0]
    ],
    "w": [
        [1, 0, 0, 1],
        [1, 0, 0, 0],
        [0, 0, 0, 1],
        [-1, 0, 1, 0],
        [0, 0, 1, 1],
        [1, 1, 0, 0],
        [0, 1, 0, -1]
    ]
}

Reduced scheme format

The reduced scheme format is used to store bilinear algorithms after additive-complexity reduction. It contains both the "fresh-variable" representation (used during common-subexpression elimination) and the final reduced linear forms.

Top-level structure

{
    "n": [n₁, n₂, n₃],
    "m": rank,
    "z2": false,
    "complexity": {"naive": x, "reduced": y},
    "u_fresh": [...],
    "v_fresh": [...],
    "w_fresh": [...],
    "u": [...],
    "v": [...],
    "w": [...]
}

Fields

n, m, z2 - these fields have the same meaning as in the full scheme format (matrix dimensions, number of bilinear multiplications and binary field flag);

Complexity:

naive - total number of additions before any reduction;
reduced - number of additions after elimination of common subexpressions and simplification.

Fresh-variable representation

The reducer may introduce fresh intermediate variables to eliminate repeated subexpressions. These are stored in three arrays: u_fresh, v_fresh and w_fresh.

Each array contains sparse linear forms written as:

[{ "index": i, "value": c }, ...]

Important indexing rule

Fresh-variable indices are allocated in consecutive blocks:

For U: original indices: 0 ... n₁·n₂ - 1, fresh indices start from: n1·n2;
For V: original indices: 0 ... n₂·n₃ - 1, fresh indices start from: n2·n3;
For W: original indices: 0 ... m - 1, fresh indices start from: m.

Thus the reducer’s intermediate variables do not collide with original matrix entries. Each list entry corresponds to one intermediate expression introduced during reduction.

Reduced linear forms

After performing additive-complexity minimization, the reducer outputs the final optimized linear forms in u, v and w. u and v arrays have exactly m rows each, w have n₃·n₁ rows, and each row represents a sparse linear form:

[{ "index": i, "value": c }, ...]

Example

Reduces 2×2×2:7 from 24 to 15 additions:

{
    "n": [2, 2, 2],
    "m": 7,
    "z2": true,
    "complexity": {"naive": 24, "reduced": 15},
    "u_fresh": [
        [{"index": 2, "value": 1}, {"index": 3, "value": 1}],
        [{"index": 1, "value": 1}, {"index": 4, "value": 1}]
    ],
    "v_fresh": [
        [{"index": 2, "value": 1}, {"index": 3, "value": 1}],
        [{"index": 1, "value": 1}, {"index": 4, "value": 1}]
    ],
    "w_fresh": [
        [{"index": 2, "value": 1}, {"index": 3, "value": 1}],
        [{"index": 0, "value": 1}, {"index": 7, "value": 1}]
    ],
    "u": [
        [{"index": 4, "value": 1}],
        [{"index": 2, "value": 1}],
        [{"index": 1, "value": 1}],
        [{"index": 5, "value": 1}],
        [{"index": 0, "value": 1}],
        [{"index": 0, "value": 1}, {"index": 5, "value": 1}],
        [{"index": 1, "value": 1}, {"index": 3, "value": 1}]
    ],
    "v": [
        [{"index": 4, "value": 1}],
        [{"index": 0, "value": 1}, {"index": 5, "value": 1}],
        [{"index": 2, "value": 1}],
        [{"index": 5, "value": 1}],
        [{"index": 0, "value": 1}],
        [{"index": 1, "value": 1}],
        [{"index": 1, "value": 1}, {"index": 3, "value": 1}]
    ],
    "w": [
        [{"index": 2, "value": 1}, {"index": 4, "value": 1}],
        [{"index": 1, "value": 1}, {"index": 6, "value": 1}, {"index": 7, "value": 1}],
        [{"index": 5, "value": 1}, {"index": 8, "value": 1}],
        [{"index": 6, "value": 1}, {"index": 8, "value": 1}]
    ]
}

Loading Schemes

The repository provides a Scheme class with a load method that supports all scheme formats used here:

Full scheme format (.json);
Addition-reduced scheme format (reduced.json);
Maple format (.m)
Plain text expressions (.exp)
Maple tensor representation (.tensor.mpl)

This allows seamless integration of schemes produced by different tools and sources.

Example usage

from src.schemes.scheme import Scheme

scheme = Scheme.load("scheme.json")
scheme.show()  # print the scheme in human-readable format
scheme.show_tensors()  # print the scheme in (a)×(b)×(c) format

# scheme saving
scheme.save("scheme.json")  # save in json format
scheme.save_maple("scheme.m")  # save in maple format
scheme.save_txt("scheme.txt")  # save in txt format
scheme.save_tensor_mpl("scheme.tensor.mpl")  # save in tensor.mpl format

Research Findings & Status

The table below summarizes the current state of researched matrix multiplication schemes. It highlights where ternary schemes (ZT) match or approximate the known minimal ranks from other fields. The best ranks of previously known schemes are given in brackets.

Format	`ZT` rank	`Z` rank	`Q` rank	ω
`2×2×2`	7	7	7	2.807354922
`2×2×3`	11	11	11	2.894952138
`2×2×4`	14	14	14	2.855516192
`2×2×5`	18	18	18	2.894489388
`2×2×6`	21	21	21	2.873949845
`2×2×7`	25	25	25	2.897969631
`2×2×8`	28	28	28	2.884412953
`2×2×9`	32	32	32	2.901396054
`2×2×10`	35	35	35	2.891404915
`2×2×11`	39	39	39	2.904369496
`2×2×12`	42	42	42	2.896519407
`2×2×13`	46	46	46	2.906913622
`2×2×14`	49	49	49	2.900482192
`2×2×15`	53	53	53	2.909104390
`2×2×16`	56	56	56	2.903677461
`2×3×3`	15	15	15	2.810763211
`2×3×4`	20	20	20	2.827893201
`2×3×5`	25	25	25	2.839184673
`2×3×6`	30	30	30	2.847366603
`2×3×7`	35	35	35	2.853661579
`2×3×8`	40	40	40	2.858709308
`2×3×9`	45	45	45	2.862881209
`2×3×10`	50 (?)	50	50	2.866409712
`2×3×11`	55	55	55	2.869448748
`2×3×12`	60	60	60	2.872104893
`2×3×13`	65 (?)	65	65	2.874454619
`2×3×14`	70	70	70	2.876554438
`2×3×15`	75 (?)	75	75	2.878447154
`2×3×16`	80	80	80	2.880165875
`2×4×4`	26	26	26	2.820263831
`2×4×5`	33 (?)	33	32	2.818527371
`2×4×6`	39 (?)	39	39	2.839089189
`2×4×7`	45	45	45	2.837016079
`2×4×8`	51	51	51	2.836212671
`2×4×9`	58 (?)	58 (?)	58	2.848323599
`2×4×10`	64 (?)	64 (?)	64	2.847232637
`2×4×11`	70 (?)	70 (?)	70 (71)	2.846666725
`2×4×12`	77 (?)	77 (?)	77	2.855044295
`2×4×13`	83 (?)	83 (?)	83	2.854307941
`2×4×14`	90	90	90	2.860958406
`2×4×15`	96 (?)	96 (?)	96	2.860170902
`2×4×16`	102	102	102	2.859610861
`2×5×5`	40	40	40	2.828878651
`2×5×6`	47	47	47	2.821072489
`2×5×7`	55 (?)	55 (?)	55	2.829707666
`2×5×8`	63 (?)	63 (?)	63	2.836451080
`2×5×9`	72 (?)	72 (?)	72	2.851231340
`2×5×10`	79 (?)	79 (?)	79	2.846440637
`2×5×11`	87	87	87	2.850288335
`2×5×12`	94	94	94	2.846978142
`2×5×13`	102 (?)	102 (?)	102	2.850502360
`2×5×14`	110 (?)	110 (?)	110	2.853593986
`2×5×15`	118 (?)	118 (?)	118	2.856335182
`2×5×16`	126 (?)	126 (?)	126	2.858787945
`2×6×6`	56 (?)	56	56	2.823707705
`2×6×7`	66 (?)	66	66	2.836714944
`2×6×8`	75 (?)	75 (?)	75	2.837746771
`2×6×9`	86 (?)	86	86	2.854051123
`2×6×10`	94	94	94	2.846978142
`2×6×11`	103 (?)	103	103	2.847583659
`2×6×12`	112 (?)	112	112	2.848295451
`2×6×13`	122 (?)	122 (?)	122	2.853955264
`2×6×14`	131 (?)	131 (?)	131	2.854351051
`2×6×15`	141	141	141	2.858926060
`2×6×16`	150 (?)	150	150	2.859138205
`2×7×7`	76 (?)	76 (?)	76	2.833651510
`2×7×8`	88 (?)	88	88	2.846670267
`2×7×9`	100 (?)	99 (?)	99	2.850404467
`2×7×10`	110 (?)	110	110	2.853593986
`2×7×11`	121 (?)	121	121	2.856364308
`2×7×12`	131 (?)	131 (?)	131	2.854351051
`2×7×13`	142 (?)	142 (?)	142	2.856929683
`2×7×14`	152 (?)	152 (?)	152	2.855497187
`2×7×15`	164 (?)	164 (?)	164	2.861285133
`2×7×16`	176 (?)	175 (?)	175	2.863150652
`2×8×8`	100	100	100	2.847366938
`2×8×9`	113 (?)	113 (?)	113	2.853661214
`2×8×10`	126 (?)	125	125	2.854077858
`2×8×11`	138 (?)	138	138	2.858873767
`2×8×12`	150 (?)	150	150	2.859138205
`2×8×13`	163 (?)	163 (?)	163 (164)	2.862977345
`2×8×14`	175 (?)	175 (?)	175	2.863150652
`2×8×15`	188 (?)	188	188	2.866331117
`2×8×16`	200	200	200	2.866446071
`2×9×9`	126	126	126	2.851807566
`2×9×10`	143 (?)	140	140	2.854814260
`2×9×11`	157 (?)	154	154	2.857430935
`2×9×12`	171 (?)	168	168	2.859738747
`2×9×13`	184 (?)	184 (?)	182	2.861796718
`2×9×14`	198 (?)	198 (?)	196	2.863648982
`2×9×15`	212 (?)	212 (?)	210	2.865329321
`2×9×16`	226 (?)	225 (?)	224	2.866864110
`2×10×10`	155	155	155	2.855675548
`2×10×11`	173 (?)	171	171	2.859854622
`2×10×12`	188 (?)	186	186	2.860476715
`2×10×13`	204 (?)	202	202	2.863822126
`2×10×14`	219 (?)	219 (?)	217	2.864293647
`2×10×15`	234 (235)	234 (235)	234 (235)	2.869317583
`2×10×16`	249	249	249	2.869528014
`2×11×11`	187	187	187	2.859082510
`2×11×12`	206 (?)	204 (?)	204	2.861281494
`2×11×13`	224 (?)	221 (?)	221	2.863244617
`2×11×14`	241 (?)	238 (?)	238	2.865013288
`2×11×15`	257 (?)	257 (?)	255	2.866619250
`2×11×16`	274 (?)	274 (?)	272	2.868087316
`2×12×12`	222	222	222	2.862112883
`2×12×13`	243 (?)	241	241	2.865119566
`2×12×14`	262 (?)	259	259	2.865766826
`2×12×15`	281 (?)	278	278	2.868257722
`2×12×16`	299 (?)	298 (?)	298 (300)	2.872173939
`2×13×13`	260	260	260	2.864831429
`2×13×14`	283 (?)	283 (?)	280	2.866530057
`2×13×15`	305 (?)	300 (?)	300	2.868073511
`2×13×16`	325 (?)	320 (?)	320	2.869485347
`2×14×14`	301	301	301	2.867288565
`2×14×15`	327 (?)	323	323	2.869573850
`2×14×16`	350 (?)	344	344	2.870191390
`2×15×15`	345	345	345	2.869524113
`2×15×16`	375 (?)	368 (?)	368	2.870888061
`2×16×16`	392	392	392	2.871569948
`3×3×3`	23	23	23	2.854049830
`3×3×4`	29	29	29	2.818985378
`3×3×5`	36	36	36	2.824142367
`3×3×6`	42 (?)	42	40	2.774299980
`3×3×7`	49 (?)	49 (?)	49	2.818025883
`3×3×8`	56 (?)	56 (?)	55	2.811068066
`3×3×9`	63 (?)	63 (?)	63	2.828432812
`3×3×10`	69 (?)	69 (?)	69	2.822857064
`3×3×11`	76 (?)	76 (?)	76	2.827390894
`3×3×12`	84 (?)	84 (?)	80	2.807712827
`3×3×13`	91 (?)	91 (?)	89	2.827681075
`3×3×14`	98 (?)	98 (?)	95	2.824820996
`3×3×15`	105 (?)	105 (?)	103	2.834537838
`3×3×16`	111 (?)	111 (?)	109	2.831905908
`3×4×4`	38	38	38	2.818959400
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`6×10×15`	594 (?)	594 (?)	594 (597)	2.816748899
`6×10×16`	630 (?)	630 (?)	630	2.815981845
`6×11×11`	496 (?)	496 (?)	490	2.820960164
`6×11×12`	521 (?)	521 (?)	521 (524)	2.811757811
`6×11×13`	584 (?)	584 (?)	574	2.821466352
`6×11×14`	621 (?)	621 (?)	613	2.819725683
`6×11×15`	653 (?)	653 (661)	653 (661)	2.819014665
`6×11×16`	700 (?)	700 (?)	684 (695)	2.812868273
`6×12×12`	564 (?)	564 (?)	560	2.807602758
`6×12×13`	624 (?)	624 (?)	616	2.816548366
`6×12×14`	666 (?)	654 (?)	645 (658)	2.806322591
`6×12×15`	704 (?)	703 (705)	686 (705)	2.805072101
`6×12×16`	736 (?)	736 (?)	736 (746)	2.809331029
`6×13×13`	682 (?)	682 (?)	678 (680)	2.825542860
`6×13×14`	730 (?)	730 (?)	726 (730)	2.824944345
`6×13×15`	763 (?)	763 (771)	763 (771)	2.818464723
`6×13×16`	819 (?)	816 (?)	816 (819)	2.821209653
`6×14×14`	777 (?)	777 (?)	776 (777)	2.823594480
`6×14×15`	814 (?)	814 (?)	814 (825)	2.816396649
`6×14×16`	864 (?)	864 (?)	864 (880)	2.815990055
`6×15×15`	868 (870)	868 (870)	859 (870)	2.811834182
`6×15×16`	920 (?)	920 (?)	920 (928)	2.815181444
`6×16×16`	972 (988)	972 (988)	972 (988)	2.812899669
`7×7×7`	250 (?)	250 (?)	249	2.835409898
`7×7×8`	278 (?)	278 (?)	277	2.825542234
`7×7×9`	316 (?)	316 (318)	315	2.834224130
`7×7×10`	346 (?)	346	345 (346)	2.830075228
`7×7×11`	378 (?)	378 (?)	376	2.828230821
`7×7×12`	404 (?)	404 (?)	402	2.821095589
`7×7×13`	443 (?)	443 (?)	441	2.829144541
`7×7×14`	475 (?)	475 (?)	471	2.827273046
`7×7×15`	511 (?)	511 (?)	508	2.832092591
`7×7×16`	540 (?)	540 (?)	539	2.831330828
`7×8×8`	310 (?)	310 (?)	306	2.812667793
`7×8×9`	347 (?)	347 (?)	347 (350)	2.820049700
`7×8×10`	385 (?)	385	385	2.822362266
`7×8×11`	423 (?)	423 (?)	423	2.824446365
`7×8×12`	452 (?)	452 (?)	452 (454)	2.817253261
`7×8×13`	498 (?)	498 (?)	496	2.825322795
`7×8×14`	532 (?)	532 (?)	529	2.822900761
`7×8×15`	572 (?)	572 (?)	557 (571)	2.816955831
`7×8×16`	598 (?)	598 (?)	598 (603)	2.821556397
`7×9×9`	396 (?)	396 (?)	396 (398)	2.830161790
`7×9×10`	433 (?)	433 (437)	433 (437)	2.825473910
`7×9×11`	478 (?)	478 (?)	478 (480)	2.829651018
`7×9×12`	513 (?)	513 (?)	508 (510)	2.820055471
`7×9×13`	563 (?)	563 (?)	562	2.831584296
`7×9×14`	600 (?)	600 (?)	597	2.827367786
`7×9×15`	639 (?)	639	634 (639)	2.825226157
`7×9×16`	677 (?)	677 (?)	667	2.820871928
`7×10×10`	478	478	478	2.825309911
`7×10×11`	526 (?)	526	526	2.827986649
`7×10×12`	564 (?)	564	557 (564)	2.816955831
`7×10×13`	614 (?)	614 (?)	614	2.826761780
`7×10×14`	653 (?)	653 (?)	653	2.823169941
`7×10×15`	694 (?)	694 (?)	694 (711)	2.821431419
`7×10×16`	742 (?)	742 (?)	736 (752)	2.820602981
`7×11×11`	580 (?)	580 (?)	577	2.829186234
`7×11×12`	624 (?)	624 (?)	618	2.823294520
`7×11×13`	680 (?)	680 (?)	675	2.828894453
`7×11×14`	725 (?)	725 (?)	721	2.827195395
`7×11×15`	778 (?)	778	777 (778)	2.831357038
`7×11×16`	822 (?)	822 (?)	822 (827)	2.829413433
`7×12×12`	669 (?)	669 (?)	660	2.816295309
`7×12×13`	731 (?)	731 (?)	724	2.823761363
`7×12×14`	780 (?)	780 (?)	774	2.822499419
`7×12×15`	815 (?)	815 (831)	815 (831)	2.816912591
`7×12×16`	884 (?)	878 (?)	878 (880)	2.822684315
`7×13×13`	798 (?)	798 (?)	794 (795)	2.830948485
`7×13×14`	852 (?)	852 (?)	850 (852)	2.830202017
`7×13×15`	909 (?)	909	909	2.831041821
`7×13×16`	966 (?)	966 (?)	962 (968)	2.829297704
`7×14×14`	909 (?)	909 (?)	909 (912)	2.829037251
`7×14×15`	952 (?)	952 (976)	952 (976)	2.821286881
`7×14×16`	1036 (?)	1036 (?)	1022 (1034)	2.825469455
`7×15×15`	1032	1032	1032	2.827727792
`7×15×16`	1089 (?)	1089 (?)	1083 (1099)	2.822639497
`7×16×16`	1164 (?)	1164 (?)	1148	2.821663673
`8×8×8`	343 (?)	343 (?)	336	2.797439141
`8×8×9`	391 (?)	391 (?)	388	2.813516852
`8×8×10`	427	427	427	2.812108880
`8×8×11`	475 (?)	475 (?)	475	2.819973988
`8×8×12`	508 (?)	508 (?)	504	2.809801266
`8×8×13`	559 (?)	559 (?)	559	2.822564153
`8×8×14`	595	595	595	2.819336895
`8×8×15`	639 (?)	639 (?)	628 (635)	2.814592730
`8×8×16`	666 (?)	666 (?)	666 (672)	2.813813510
`8×9×9`	432 (?)	432 (?)	430	2.809957177
`8×9×10`	482 (487)	482 (487)	482 (487)	2.817012414
`8×9×11`	521 (?)	521 (?)	521 (533)	2.811757811
`8×9×12`	564 (?)	564 (?)	560	2.807602758
`8×9×13`	617 (?)	617 (624)	617 (624)	2.817259628
`8×9×14`	654 (?)	654 (669)	654 (669)	2.812333691
`8×9×15`	699 (?)	699 (705)	699 (705)	2.813135327
`8×9×16`	735 (?)	735 (?)	735 (746)	2.808752406
`8×10×10`	532	532	532	2.816907135
`8×10×11`	588 (?)	588	588	2.821593096
`8×10×12`	630 (?)	630	624 (630)	2.811801179
`8×10×13`	686 (?)	686	686	2.820311011
`8×10×14`	728 (?)	728	728	2.815933159
`8×10×15`	784 (?)	784 (789)	778 (789)	2.816637962
`8×10×16`	826 (?)	826 (?)	822 (832)	2.814298209
`8×11×11`	646 (?)	646 (?)	641	2.820135833
`8×11×12`	690 (?)	690 (?)	676 (680)	2.807798855
`8×11×13`	754 (?)	754 (?)	750	2.820138111
`8×11×14`	804 (?)	804	804	2.820079580
`8×11×15`	859 (?)	859	848 (859)	2.815247371
`8×11×16`	914 (?)	914 (?)	904 (920)	2.816647975
`8×12×12`	735 (?)	735 (?)	720	2.799977314
`8×12×13`	807 (?)	807 (?)	784 (798)	2.804375455
`8×12×14`	861 (?)	861	843 (861)	2.805742480
`8×12×15`	914 (915)	914 (915)	904 (915)	2.807944089
`8×12×16`	960 (?)	960 (?)	960	2.807820225
`8×13×13`	885 (?)	885 (?)	880	2.821307498
`8×13×14`	945 (?)	945	945	2.821953852
`8×13×15`	992 (1005)	992 (1005)	992 (1005)	2.815278496
`8×13×16`	1072 (?)	1072 (?)	1054 (1064)	2.815302758
`8×14×14`	1008 (?)	1008	1004 (1008)	2.818224059
`8×14×15`	1063 (1080)	1063 (1080)	1063 (1080)	2.815109808
`8×14×16`	1127 (?)	1127 (?)	1104 (1138)	2.806012534
`8×15×15`	1137 (1140)	1137 (1140)	1130 (1140)	2.813661626
`8×15×16`	1204 (?)	1204 (?)	1185 (1198)	2.808501081
`8×16×16`	1260 (?)	1260 (?)	1230 (1248)	2.799393436
`9×9×9`	486 (498)	486 (498)	486 (498)	2.815464877
`9×9×10`	537 (?)	537 (?)	534	2.813362874
`9×9×11`	594 (?)	594 (?)	576	2.807325686
`9×9×12`	626 (?)	626 (?)	600	2.789620062
`9×9×13`	683 (?)	683 (?)	681	2.812123330
`9×9×14`	725 (?)	725 (?)	720 (726)	2.806246597
`9×9×15`	794 (?)	794 (?)	760 (783)	2.801824302
`9×9×16`	824 (?)	824 (?)	823 (825)	2.809929143
`9×10×10`	597 (?)	597 (600)	597 (600)	2.818970672
`9×10×11`	661 (?)	661 (?)	651	2.817680531
`9×10×12`	702 (?)	702 (?)	668 (684)	2.793651686
`9×10×13`	771 (?)	763 (772)	758 (772)	2.815672846
`9×10×14`	820 (?)	820	808 (820)	2.813287623
`9×10×15`	870	865 (870)	865 (870)	2.814731263
`9×10×16`	924 (?)	920 (?)	916 (939)	2.813383975
`9×11×11`	721 (?)	721 (?)	715 (725)	2.819505933
`9×11×12`	762 (?)	762 (?)	738 (760)	2.798270808
`9×11×13`	843 (?)	843 (849)	835 (849)	2.818729064
`9×11×14`	900 (?)	900 (904)	882 (904)	2.812562599
`9×11×15`	958 (?)	958 (?)	958 (981)	2.819945748
`9×11×16`	1024 (?)	996 (1030)	996 (1030)	2.811083198
`9×12×12`	810 (?)	810 (?)	800	2.798064630
`9×12×13`	894 (?)	894 (?)	878 (900)	2.805673201
`9×12×14`	960 (?)	960 (?)	940 (945)	2.805232985
`9×12×15`	1012 (?)	1012 (?)	996 (1000)	2.802534955
`9×12×16`	1074 (?)	1074 (?)	1035 (1080)	2.793729377
`9×13×13`	986 (?)	986 (996)	981 (996)	2.820440786
`9×13×14`	1050 (?)	1050 (1063)	1026 (1063)	2.810379562
`9×13×15`	1119 (?)	1119 (?)	1119 (1135)	2.819269106
`9×13×16`	1188 (?)	1188 (1210)	1179 (1210)	2.815916926
`9×14×14`	1125 (?)	1125 (1136)	1101 (1136)	2.810831956
`9×14×15`	1179 (?)	1179 (?)	1175 (1185)	2.810993734
`9×14×16`	1270 (?)	1270 (?)	1254 (1260)	2.812806553
`9×15×15`	1276 (?)	1276 (?)	1236 (1290)	2.805463706
`9×15×16`	1320 (?)	1320 (1350)	1320 (1350)	2.807572842
`9×16×16`	1380 (1444)	1380 (1444)	1380 (1444)	2.801393711
`10×10×10`	651 (?)	651	651	2.813580989
`10×10×11`	719 (?)	719	719	2.817849439
`10×10×12`	768 (?)	768 (770)	768 (770)	2.811164062
`10×10×13`	838 (?)	838	838	2.816278610
`10×10×14`	889 (?)	889	889	2.811934283
`10×10×15`	957 (?)	957 (?)	957	2.815641959
`10×10×16`	1008 (?)	1008 (?)	1008	2.812123655
`10×11×11`	793 (?)	793	793	2.821415868
`10×11×12`	850 (?)	850	849 (850)	2.815739433
`10×11×13`	924 (?)	924	924	2.819673026
`10×11×14`	981 (?)	981	981	2.815670174
`10×11×15`	1050 (?)	1050 (?)	1050 (1067)	2.816973777
`10×11×16`	1112 (?)	1112 (?)	1112 (1136)	2.815676689
`10×12×12`	902 (?)	902 (910)	902 (910)	2.807030426
`10×12×13`	990 (?)	990	990	2.814455029
`10×12×14`	1050 (?)	1050	1050	2.810139154
`10×12×15`	1122 (1140)	1122 (1140)	1122 (1140)	2.810818006
`10×12×16`	1190 (?)	1190 (?)	1176 (1216)	2.805475746
`10×13×13`	1082 (?)	1082	1082	2.820012787
`10×13×14`	1154 (?)	1154	1154	2.817919098
`10×13×15`	1230 (?)	1230 (1242)	1230 (1242)	2.817513014
`10×13×16`	1326 (?)	1326 (1332)	1318 (1332)	2.820846164
`10×14×14`	1232 (?)	1232	1232	2.816254849
`10×14×15`	1316 (?)	1314 (1327)	1314 (1327)	2.816125387
`10×14×16`	1418 (?)	1418 (1423)	1398 (1423)	2.816663561
`10×15×15`	1395 (?)	1389 (1395)	1385 (1395)	2.811406977
`10×15×16`	1488 (?)	1484 (1497)	1482 (1497)	2.814186431
`10×16×16`	1585 (1586)	1578 (1586)	1560 (1586)	2.810651214
`11×11×11`	873 (?)	873	873	2.824116479
`11×11×12`	936 (?)	936	922 (936)	2.812867384
`11×11×13`	1023 (?)	1023	1023	2.824645969
`11×11×14`	1093 (?)	1093	1093	2.823197571
`11×11×15`	1169 (?)	1169 (1181)	1169 (1181)	2.824115356
`11×11×16`	1230 (?)	1230 (?)	1230 (1236)	2.820195393
`11×12×12`	1002 (?)	1002 (?)	968 (990)	2.799472327
`11×12×13`	1102 (?)	1102	1082 (1102)	2.814231919
`11×12×14`	1182 (?)	1182	1153 (1182)	2.811853672
`11×12×15`	1248 (?)	1234 (?)	1234 (1264)	2.813129312
`11×12×16`	1306 (?)	1306 (?)	1278 (1312)	2.803143027
`11×13×13`	1205 (?)	1205 (1210)	1205 (1210)	2.827216655
`11×13×14`	1292 (?)	1292 (1298)	1292 (1298)	2.827166171
`11×13×15`	1377 (?)	1377	1371 (1377)	2.824949046
`11×13×16`	1458 (?)	1458 (1472)	1446 (1472)	2.822035717
`11×14×14`	1376 (?)	1376 (1388)	1376 (1388)	2.824489318
`11×14×15`	1432 (?)	1432 (1471)	1432 (1471)	2.814780394
`11×14×16`	1550 (?)	1550 (?)	1520 (1571)	2.814428649
`11×15×15`	1548 (?)	1548 (?)	1540	2.817843009
`11×15×16`	1657 (?)	1629 (?)	1605 (1656)	2.810502126
`11×16×16`	1752 (?)	1752 (?)	1724	2.814679929
`12×12×12`	1068 (?)	1068 (?)	1040	2.795668800
`12×12×13`	1168 (?)	1168 (?)	1144 (1152)	2.803918249
`12×12×14`	1260 (?)	1260 (?)	1234 (1250)	2.806467563
`12×12×15`	1332 (?)	1332 (?)	1280	2.795549318
`12×12×16`	1380 (?)	1380 (?)	1380 (1392)	2.801393711
`12×13×13`	1298 (?)	1298 (?)	1274	2.816848164
`12×13×14`	1389 (?)	1389 (?)	1370 (1382)	2.818044255
`12×13×15`	1470 (?)	1470 (?)	1442 (1460)	2.812789736
`12×13×16`	1548 (?)	1544 (?)	1544 (1556)	2.815794289
`12×14×14`	1484 (?)	1484 (?)	1449 (1481)	2.812807792
`12×14×15`	1546 (?)	1546 (?)	1538 (1540)	2.810862435
`12×14×16`	1663 (?)	1663 (?)	1617 (1638)	2.806918970
`12×15×15`	1650 (?)	1650 (?)	1600	2.801323500
`12×15×16`	1725 (?)	1725 (?)	1725 (1728)	2.806957387
`12×16×16`	1862 (?)	1862 (?)	1815 (1824)	2.803398069
`13×13×13`	1426 (?)	1426 (?)	1421 (1426)	2.830120644
`13×13×14`	1511 (?)	1511 (1524)	1511 (1524)	2.826838093
`13×13×15`	1605 (?)	1605	1605	2.825055042
`13×13×16`	1711 (?)	1711 (?)	1704 (1713)	2.824705676
`13×14×14`	1614 (?)	1614 (1625)	1614 (1625)	2.825351482
`13×14×15`	1681 (?)	1681 (1714)	1681 (1714)	2.816136526
`13×14×16`	1820 (?)	1820 (?)	1800 (1825)	2.819075643
`13×15×15`	1803 (?)	1797 (1803)	1797 (1803)	2.816875265
`13×15×16`	1908 (?)	1908 (1932)	1908 (1932)	2.816628414
`13×16×16`	2038 (?)	2038 (?)	2022	2.815680662
`14×14×14`	1725 (?)	1725 (?)	1719	2.822787486
`14×14×15`	1798 (?)	1798 (1813)	1798 (1813)	2.815280055
`14×14×16`	1943 (?)	1943 (?)	1931 (1939)	2.819303950
`14×15×15`	1905 (?)	1895 (1905)	1890 (1905)	2.809752096
`14×15×16`	2043 (?)	2043 (?)	2016	2.811264261
`14×16×16`	2170 (?)	2170 (?)	2128 (2142)	2.808914234
`15×15×15`	2058 (?)	2058 (?)	2058	2.817336958
`15×15×16`	2132 (?)	2132 (?)	2132 (2173)	2.808074285
`15×16×16`	2302 (?)	2302 (?)	2262	2.807630537
`16×16×16`	2401 (?)	2401 (?)	2304	2.792481250

Coefficient set status

total schemes: 680 (51 better Strassen)
ZT schemes: 362 (53.24%)
Z schemes: 30 (4.41%)
Q schemes: 288 (42.35%)

License and Citation

This project is for research purposes. Please use the following citation when referencing this code or dataset in your academic work:

@article{perminov2025fast,
    title={Fast Matrix Multiplication via Ternary Meta Flip Graphs},
    author={Perminov, Andrew I},
    journal={arXiv preprint arXiv:2511.20317},
    url={https://arxiv.org/abs/2511.20317},
    year={2025}
}

@article{perminov2025parallel,
    title={Parallel Heuristic Exploration for Additive Complexity Reduction in Fast Matrix Multiplication},
    author={Perminov, Andrew I},
    journal={arXiv preprint arXiv:2512.13365},
    url={https://arxiv.org/abs/2512.13365},
    year={2025}
}

@article{perminov202558,
    title={A 58-Addition, Rank-23 Scheme for General 3x3 Matrix Multiplication},
    author={Perminov, Andrew I},
    journal={arXiv preprint arXiv:2512.21980},
    url={https://arxiv.org/abs/2512.21980},
    year={2025}
}

@article{perminov2026fast,
    title={Fast Matrix Multiplication in Small Formats: Discovering New Schemes with an Open-Source Flip Graph Framework},
    author={Perminov, Andrew I},
    journal={arXiv preprint arXiv:2603.02398},
    url={https://arxiv.org/abs/2603.02398},
    year={2026}
}

Name		Name	Last commit message	Last commit date
Latest commit History 302 Commits
schemes		schemes
scripts		scripts
src		src
.gitignore		.gitignore
CHANGELOG.md		CHANGELOG.md
LICENSE		LICENSE
README.md		README.md
find_alternative_schemes.py		find_alternative_schemes.py
lift_schemes.py		lift_schemes.py
requirements.txt		requirements.txt
update_schemes_status.py		update_schemes_status.py

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FastMatrixMultiplication

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Rediscovery in the ternary coefficient set (ZT)

Rediscovery in the integer ring (Z)

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ternary_flip_graph: core flip graph exploration toolkit

ternary_addition_reducer: addition reduction tool

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Rediscovery in the ternary coefficient set (`ZT`)

Rediscovery in the integer ring (`Z`)

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