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

• Fast Matrix Multiplication via Ternary Meta Flip Graphs (arxiv)
• Parallel Heuristic Exploration for Additive Complexity Reduction in Fast Matrix Multiplication (arxiv)
• A 58-Addition, Rank-23 Scheme for General 3x3 Matrix Multiplication (arxiv)
• Fast Matrix Multiplication in Small Formats: Discovering New Schemes with an Open-Source Flip Graph Framework (arxiv)

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
3×5×9	104 (Z)	102 (ZT)	2.828571093
3×5×10	115 (Z)	114 (ZT)	2.835687395
3×9×11	224 (Q)	222 (Q)	2.846644652
3×10×11	249 (Q)	248 (Q)	2.852219687
4×4×10	120 (Q)	115 (ZT)	2.804789925
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
4×7×8	164 (ZT)	163 (Z)	2.823771262
4×7×11	227 (Z)	226 (ZT)	2.837927019
4×9×11	280 (ZT)	279 (ZT)	2.824354701
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)	208 (ZT)	2.807367621
5×5×13	237 (Z)	228 (ZT)	2.816135753
5×5×14	254 (Z)	248 (ZT)	2.823570328
5×5×15	271 (Q)	266 (ZT)	2.826168043
5×5×16	288 (Q)	284 (ZT)	2.828510889
5×6×9	197 (Z)	193 (ZT)	2.820092998
5×6×10	218 (Z)	217 (ZT)	2.829647192
5×7×8	205 (Q)	204 (ZT)	2.831402964
5×9×9	294 (Q)	293 (ZT)	2.838247561
6×7×7	215 (ZT)	212 (ZT)	2.827400948
6×7×8	239 (ZT)	238 (ZT)	2.822158898
6×7×9	270 (ZT)	268 (ZT)	2.826160061
6×7×10	296 (Z)	293 (Q)	2.821158816
7×7×10	346 (Z)	345 (Q)	2.830075228
7×8×15	571 (Q)	570 (Q)	2.827234938
7×9×11	480 (Q)	478 (ZT)	2.829651018
7×9×15	639 (Z)	634 (Q)	2.825226157
7×13×16	968 (Q)	966 (Q)	2.831006805
7×14×15	976 (Z)	969 (ZT/Z)	2.828567644
7×14×16	1034 (Q)	1030 (Q)	2.828648763
8×8×16	672 (Q)	671 (Q)	2.817050687
8×9×11	533 (Q)	531 (ZT)	2.820303082
8×9×14	669 (Z)	666 (ZT)	2.820221074
9×10×10	600 (Z)	597 (ZT)	2.818970672
9×10×13	772 (Z)	765 (Q)	2.819576354
9×10×14	820 (Z)	819 (Q)	2.818970048
9×11×11	725 (Q)	715 (Q)	2.819505933
9×11×12	760 (Q)	754 (Q)	2.807359140
9×11×13	849 (Z)	835 (Q)	2.818729064
9×11×14	904 (Z)	889 (Q)	2.815840862
9×11×15	981 (Q)	960 (ZT/Z)	2.820802434
9×11×16	1030 (Z)	1023 (Q)	2.821974308
9×12×13	900 (Q)	884 (Q)	2.808492461
9×12×16	1080 (Q)	1072 (Q)	2.807864570
9×13×13	996 (Z)	981 (Q)	2.820440786
9×13×14	1063 (Z)	1041 (Q)	2.816262654
9×13×15	1135 (Q)	1119 (ZT/Z)	2.819269106
9×13×16	1210 (Z)	1183 (Q)	2.817265461
9×14×14	1136 (Z)	1121 (Q)	2.818056626
9×15×15	1290 (Q)	1284 (ZT/Z)	2.820476797
9×15×16	1350 (Z)	1341 (Q)	2.813740131
9×16×16	1444 (ZT)	1431 (Q)	2.815455240
10×10×12	770 (Z)	768 (ZT)	2.811164062
10×11×15	1067 (Q)	1055 (ZT)	2.818897479
10×13×16	1332 (Z)	1326 (ZT)	2.823222352
11×11×15	1170 (Z)	1169 (ZT)	2.824115356
11×12×13	1102 (Z)	1092 (Q)	2.817937614
11×12×15	1264 (Q)	1240 (Q)	2.815046272
11×13×13	1210 (Z)	1205 (ZT)	2.827216655
11×13×14	1298 (Z)	1292 (ZT)	2.827166171
11×13×16	1472 (Z)	1452 (Q)	2.823641623
11×14×14	1388 (Z)	1376 (ZT)	2.824489318
11×14×15	1471 (Z)	1460 (ZT/Z)	2.822281114
11×14×16	1571 (Q)	1548 (Q)	2.821440626
12×12×14	1250 (Q)	1240 (Q)	2.808379984
12×13×16	1556 (Q)	1548 (ZT/Q)	2.816786558
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)	1698 (ZT)	2.819951805
13×14×16	1825 (Q)	1806 (Q)	2.820327226
13×15×16	1932 (Z)	1908 (Q)	2.816628414
14×14×16	1939 (Q)	1938 (Q)	2.820652346
15×15×16	2173 (Q)	2155 (Q)	2.812005396

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
2×4×11	71	Q
2×4×12	77	Q
2×4×15	96	Q
2×5×7	55	Q
2×5×8	63	Q
2×5×9	72	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
2×6×9	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
2×7×8	88	Z
2×7×10	110	Z
2×7×11	121	Z
2×8×15	188	Z
3×3×7	49	Q
3×3×9	63	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
3×10×16	360	Q
4×4×6	73	Z/Q
4×4×8	96	Q
4×4×11	130	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
4×7×12	246	Z
4×7×15	307	Q
4×8×13	297	Z
4×9×14	355	Z
4×9×15	375	Z
4×10×13	361	Q
4×10×14	385	Q
4×10×15	417	Q
4×10×16	441	Q
4×11×11	340	Z
4×11×12	365	Z
4×11×14	429	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
5×9×15	474	Z
5×10×11	386	Z
5×10×12	413	Z
5×10×13	451	Q
5×10×14	481	Q
5×10×15	519	Q
5×10×16	549	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
6×8×16	511	Q
6×9×9	342	Z
6×9×10	373	Z
6×9×11	407	Q
6×9×12	434	Q
6×10×11	446	Z
6×10×12	476	Z
6×10×13	520	Q
6×10×14	553	Q
6×10×15	597	Q
6×10×16	630	Q
6×11×15	661	Z
6×12×15	705	Z
6×12×16	746	Q
6×13×15	771	Z
6×13×16	819	Q
6×14×14	777	Q
6×14×15	825	Q
6×14×16	880	Q
7×8×10	385	Z
7×8×11	423	Q
7×8×12	454	Q
7×8×16	603	Q
7×9×10	437	Z
7×10×11	526	Z
7×10×12	564	Z
7×10×13	614	Q
7×10×14	653	Q
7×10×15	703	Q
7×10×16	742	Q
7×11×16	822	Q
7×12×15	831	Z
7×13×15	909	Z
8×8×11	475	Q
8×8×13	559	Q
8×9×13	624	Z
8×9×15	705	Z
8×9×16	746	Q
8×10×11	588	Z
8×10×12	630	Z
8×10×13	686	Z
8×10×14	728	Z
8×11×14	804	Z
8×11×15	859	Z
8×12×14	861	Z
8×13×14	945	Z
8×14×14	1008	Z
9×14×15	1185	Q
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
10×11×12	850	Z
10×11×13	924	Z
10×11×14	981	Z
10×12×12	910	Z
10×12×13	990	Z
10×12×14	1050	Z
10×13×13	1082	Z
10×13×14	1154	Z
10×13×15	1242	Z
10×14×14	1232	Z
10×14×15	1327	Z
10×14×16	1423	Z
10×15×15	1395	Z
10×15×16	1497	Z
11×11×11	873	Z
11×11×12	936	Z
11×11×13	1023	Z
11×11×14	1093	Z
11×12×14	1182	Z
11×13×15	1377	Z
13×13×15	1605	Z
13×15×15	1803	Z
14×14×15	1813	Z
14×15×15	1905	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
2×6×8	75
2×7×7	76
2×7×9	99
2×7×12	131
2×7×13	142
2×7×14	152
2×7×15	164
2×7×16	175
2×8×9	113
2×8×14	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
4×11×16	489
4×14×14	532
5×11×16	609

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

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	59 (?)	59 (?)	58	2.848323599
2×4×10	65 (?)	65 (?)	64	2.847232637
2×4×11	71 (?)	71 (?)	71	2.856171014
2×4×12	77 (?)	77 (?)	77	2.855044295
2×4×13	84 (?)	84 (?)	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	80 (?)	80 (?)	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	77 (?)	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	77 (?)	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	132 (?)	131 (?)	131	2.854351051
2×7×13	143 (?)	142 (?)	142	2.856929683
2×7×14	154 (?)	152 (?)	152	2.855497187
2×7×15	165 (?)	164 (?)	164	2.861285133
2×7×16	176 (?)	175 (?)	175	2.863150652
2×8×8	100	100	100	2.847366938
2×8×9	114 (?)	113 (?)	113	2.853661214
2×8×10	126 (?)	125	125	2.854077858
2×8×11	139 (?)	138	138	2.858873767
2×8×12	151 (?)	150	150	2.859138205
2×8×13	163 (?)	163 (?)	163 (164)	2.862977345
2×8×14	176 (?)	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	144 (?)	140	140	2.854814260
2×9×11	158 (?)	154	154	2.857430935
2×9×12	171 (?)	168	168	2.859738747
2×9×13	185 (?)	185 (?)	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	174 (?)	171	171	2.859854622
2×10×12	188 (?)	186	186	2.860476715
2×10×13	204 (?)	202	202	2.863822126
2×10×14	220 (?)	219 (?)	217	2.864293647
2×10×15	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	258 (?)	257 (?)	255	2.866619250
2×11×16	274 (?)	274 (?)	272	2.868087316
2×12×12	222	222	222	2.862112883
2×12×13	244 (?)	241	241	2.865119566
2×12×14	262 (?)	259	259	2.865766826
2×12×15	282 (?)	278	278	2.868257722
2×12×16	299 (?)	298 (?)	298 (300)	2.872173939
2×13×13	260	260	260	2.864831429
2×13×14	285 (?)	283 (?)	280	2.866530057
2×13×15	306 (?)	300 (?)	300	2.868073511
2×13×16	325 (?)	320 (?)	320	2.869485347
2×14×14	301	301	301	2.867288565
2×14×15	329 (?)	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	71 (?)	71 (?)	69	2.822857064
3×3×11	78 (?)	78 (?)	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	112 (?)	112 (?)	109	2.831905908
3×4×4	38	38	38	2.818959400
3×4×5	47 (?)	47	47	2.821072489
3×4×6	54 (?)	54	54	2.798196494
3×4×7	64 (?)	64	63	2.805217355
3×4×8	73 (74)	73 (74)	73	2.819981689
3×4×9	83 (?)	83 (?)	83	2.831300786
3×4×10	92 (?)	92 (?)	92	2.833501646
3×4×11	101 (?)	101 (?)	101	2.835536188
3×4×12	108 (?)	108 (?)	108	2.826342326
3×4×13	118 (?)	118 (?)	117	2.829094886
3×4×14	127 (?)	127 (?)	126	2.831566689
3×4×15	137 (?)	137 (?)	136	2.838067999
3×4×16	146 (?)	146 (?)	146	2.843715269
3×5×5	58	58	58	2.821392604
3×5×6	68 (?)	68	68	2.813124119
3×5×7	79 (?)	79 (80)	79	2.816599750
3×5×8	90 (?)	90	90	2.819728939
3×5×9	102 (?)	102 (104)	102 (104)	2.828571093
3×5×10	114 (?)	114 (115)	114 (115)	2.835687395
3×5×11	126 (?)	126	126	2.841559080
3×5×12	136 (?)	136	136	2.838067999
3×5×13	147 (?)	147 (?)	147	2.839237439
3×5×14	158 (?)	158 (?)	158	2.840373980
3×5×15	169 (?)	169 (?)	169	2.841471724
3×5×16	180 (?)	180	180	2.842528174
3×6×6	83 (?)	83 (?)	80	2.807712827
3×6×7	96 (?)	96 (?)	94	2.818256795
3×6×8	108 (?)	108	108	2.826342326
3×6×9	122 (?)	122 (?)	120	2.823037498
3×6×10	136 (?)	136 (?)	134	2.829509241
3×6×11	150 (?)	150 (?)	148	2.834886501
3×6×12	162 (?)	162 (?)	160	2.832508438
3×6×13	176 (?)	176 (?)	174	2.837076970
3×6×14	190 (?)	190 (?)	188	2.841039398
3×6×15	204 (?)	204 (?)	200	2.839184366
3×6×16	216 (?)	216 (?)	214	2.842670031
3×7×7	111 (?)	111 (?)	111	2.831135449
3×7×8	128 (?)	128 (?)	126	2.831566689
3×7×9	143 (?)	143 (?)	142	2.836359925
3×7×10	158 (?)	158 (?)	157	2.836811740
3×7×11	175 (?)	175 (?)	173	2.840626282
3×7×12	190 (?)	190 (?)	188	2.841039398
3×7×13	207 (?)	207 (?)	205	2.846797438
3×7×14	222 (?)	222 (?)	220	2.846952678
3×7×15	237 (?)	237 (?)	236	2.849419980
3×7×16	254 (?)	254 (?)	251	2.849586051
3×8×8	146 (?)	146 (?)	145	2.839793508
3×8×9	163 (?)	163 (?)	163	2.842876116
3×8×10	180 (?)	180	180	2.842528174
3×8×11	198 (?)	198 (?)	198	2.845219854
3×8×12	216 (?)	216 (?)	216	2.847598050
3×8×13	236 (?)	236 (?)	234	2.849722142
3×8×14	253 (?)	253 (?)	252	2.851636630
3×8×15	270 (?)	270	270	2.853375643
3×8×16	288 (?)	288 (?)	288	2.854965878
3×9×9	185 (?)	185 (?)	183	2.845127188
3×9×10	204 (?)	204 (?)	203	2.847162657
3×9×11	224 (?)	224 (?)	222 (224)	2.846644652
3×9×12	244 (?)	244 (?)	240	2.844256405
3×9×13	265 (?)	265 (?)	262	2.850305893
3×9×14	285 (?)	285 (?)	283	2.853688732
3×9×15	306 (?)	306 (?)	303	2.855016796
3×9×16	326 (?)	326 (?)	323	2.856252700
3×10×10	228 (?)	228 (?)	226	2.851021242
3×10×11	250 (?)	250 (?)	248 (249)	2.852219687
3×10×12	270 (?)	270 (?)	268	2.849586221
3×10×13	294 (?)	294 (?)	291	2.852757491
3×10×14	316 (?)	316 (?)	314	2.855538348
3×10×15	338 (?)	338 (?)	336	2.856543829
3×10×16	360 (?)	360 (?)	360	2.860207950
3×11×11	276 (?)	276 (?)	274	2.856843143
3×11×12	298 (?)	298 (?)	296	2.854020431
3×11×13	323 (?)	323 (?)	321	2.856462333
3×11×14	348 (?)	348 (?)	346	2.858631053
3×11×15	373 (?)	373 (?)	369	2.857961939
3×11×16	396 (?)	396 (?)	394	2.859910251
3×12×12	324 (?)	324 (?)	320	2.851639645
3×12×13	352 (?)	352 (?)	348	2.855444087
3×12×14	378 (?)	378 (?)	376	2.858746388
3×12×15	406 (?)	406 (?)	400	2.856901552
3×12×16	432 (?)	432 (?)	428	2.859827202
3×13×13	383 (?)	383 (?)	379	2.859850228
3×13×14	411 (?)	411 (?)	408	2.861318698
3×13×15	439 (?)	439 (?)	436	2.861587800
3×13×16	468 (?)	468 (?)	465	2.862908907
3×14×14	443 (?)	443 (?)	440	2.863588849
3×14×15	474 (?)	474 (?)	470	2.863636429
3×14×16	504 (?)	504 (?)	502	2.865600623
3×15×15	507 (?)	507 (?)	503	2.864557717
3×15×16	540 (?)	540 (?)	536	2.865432840
3×16×16	576 (?)	576 (?)	574	2.868526693
4×4×4	49	49	48	2.792481250
4×4×5	61	61	61	2.814364818
4×4×6	73 (?)	73	73	2.819981689
4×4×7	85	85	85	2.824617348
4×4×8	96 (?)	96 (?)	96	2.822126786
4×4×9	107 (?)	107 (?)	104	2.803560588
4×4×10	115 (?)	115 (?)	115 (120)	2.804789925
4×4×11	130 (?)	130 (?)	130	2.824223683
4×4×12	141 (?)	141 (?)	141 (142)	2.823831239
4×4×13	153 (?)	153 (?)	152	2.823706611
4×4×14	164 (?)	164 (?)	163 (165)	2.823771262
4×4×15	176 (?)	176 (?)	176 (177)	2.830226950
4×4×16	188 (?)	188 (?)	188 (189)	2.832970819
4×5×5	76	76	76	2.821220388
4×5×6	90 (?)	90	90	2.819728939
4×5×7	104 (?)	104	104	2.819542878
4×5×8	118 (122)	118	118	2.820012554
4×5×9	132 (?)	132 (139)	132 (136)	2.820821776
4×5×10	146 (152)	146 (151)	146 (151)	2.821805270
4×5×11	160 (?)	160 (165)	160 (165)	2.822872235
4×5×12	174 (?)	174 (180)	174 (180)	2.823971094
4×5×13	191 (?)	191 (194)	191 (194)	2.833613095
4×5×14	207 (?)	207 (208)	206 (208)	2.836597217
4×5×15	221 (?)	221 (226)	221 (226)	2.839254157
4×5×16	235 (?)	235 (?)	235 (240)	2.839432229
4×6×6	105	105	105	2.809337134
4×6×7	123 (?)	123	123	2.817457953
4×6×8	140	140	140	2.819769913
4×6×9	159 (?)	159 (?)	159	2.829009300
4×6×10	175 (?)	175	175	2.827107959
4×6×11	194 (?)	194 (?)	194	2.834239371
4×6×12	210	210	210	2.832674296
4×6×13	227 (?)	227 (228)	227 (228)	2.833857047
4×6×14	245	245	245	2.837108348
4×6×15	263 (?)	263	263	2.839987538
4×6×16	280	280	280	2.840763608
4×7×7	144 (?)	144	144	2.824766202
4×7×8	164	163 (164)	163 (164)	2.823771262
4×7×9	187 (?)	187 (?)	186	2.835236653
4×7×10	206 (?)	206 (?)	203	2.828786709
4×7×11	226 (?)	226 (227)	226 (227)	2.837927019
4×7×12	246 (?)	246	246	2.839209043
4×7×13	267 (?)	267 (?)	266	2.840436133
4×7×14	285	285	285	2.839846584
4×7×15	307 (?)	307 (?)	307	2.844340854
4×7×16	324	324	324	2.840756417
4×8×8	182	182	182	2.815422990
4×8×9	209 (?)	209 (?)	206	2.822486324
4×8×10	230 (?)	230 (?)	224	2.814499777
4×8×11	255 (?)	255 (?)	252	2.829012734
4×8×12	272	272	272	2.826149622
4×8×13	297 (?)	297	297	2.832380680
4×8×14	315	315	315	2.826912765
4×8×15	339	339	339	2.831001921
4×8×16	357	357	357	2.826593421
4×9×9	225	225	225	2.810763211
4×9×10	255	255	255	2.824243429
4×9×11	279 (280)	279 (280)	279 (280)	2.824354701
4×9×12	300	300	300	2.819734242
4×9×13	330 (?)	330 (?)	329	2.828049591
4×9×14	355 (?)	355	355	2.831038562
4×9×15	375 (?)	375	375	2.826127741
4×9×16	400	400	400	2.827893201
4×10×10	280	280	280	2.821408468
4×10×11	308	308	308	2.824204956
4×10×12	329	329	329	2.816452167
4×10×13	361 (?)	361 (?)	361	2.824930840
4×10×14	385 (?)	385 (?)	385	2.822362266
4×10×15	417 (?)	417 (?)	417	2.829366531
4×10×16	441 (?)	441 (?)	441	2.827087301
4×11×11	340 (?)	340	340	2.828630974
4×11×12	365 (?)	365	365	2.823324329
4×11×13	401 (?)	401 (?)	400	2.830997032
4×11×14	429 (?)	429 (?)	429	2.831024692
4×11×15	463 (?)	463 (?)	452	2.825072241
4×11×16	490 (?)	489 (?)	489	2.833264518
4×12×12	390	390	390	2.815943526
4×12×13	430 (?)	430 (?)	426	2.822077950
4×12×14	462 (?)	462 (?)	456	2.821313298
4×12×15	495 (?)	495 (?)	480	2.815116449
4×12×16	520	520	520	2.823913337
4×13×13	472 (?)	472 (?)	466	2.828730930
4×13×14	506 (?)	502 (?)	500	2.828979156
4×13×15	537 (?)	537 (?)	528	2.824216663
4×13×16	568	568	568	2.829690422
4×14×14	539 (?)	532 (?)	532	2.825446399
4×14×15	572 (?)	572 (?)	568	2.825668891
4×14×16	610	610	610	2.830324437
4×15×15	600	600	600	2.821181309
4×15×16	642 (?)	642 (?)	640	2.822861928
4×16×16	676	676	676	2.820263831
5×5×5	93	93	93	2.816262409
5×5×6	110 (?)	110	110	2.814302034
5×5×7	127 (134)	127	127	2.813778022
5×5×8	144 (?)	144	144	2.813995249
5×5×9	161 (?)	161 (167)	161 (167)	2.814610506
5×5×10	178 (?)	178 (?)	178 (184)	2.815441580
5×5×11	195 (?)	195 (?)	195 (202)	2.816386568
5×5×12	208 (?)	208 (220)	208 (220)	2.807367621
5×5×13	228 (?)	228 (237)	228 (237)	2.816135753
5×5×14	248 (?)	248 (254)	248 (254)	2.823570328
5×5×15	266 (?)	266 (?)	266 (271)	2.826168043
5×5×16	284 (?)	284 (?)	284 (288)	2.828510889
5×6×6	130 (?)	130	130	2.812001673
5×6×7	150 (?)	150	150	2.811221917
5×6×8	170 (176)	170	170	2.811240720
5×6×9	193 (?)	193 (197)	193 (197)	2.820092998
5×6×10	217 (?)	217 (218)	217 (218)	2.829647192
5×6×11	238 (?)	238 (?)	236	2.826562083
5×6×12	258 (?)	258 (?)	250	2.814150526
5×6×13	280 (?)	280 (?)	278	2.829776752
5×6×14	300 (?)	300 (?)	297	2.827893397
5×6×15	320 (?)	320 (?)	318	2.829506239
5×6×16	340 (?)	340 (?)	340	2.832433220
5×7×7	176 (?)	176	176	2.819618966
5×7×8	204 (?)	204 (?)	204 (205)	2.831402964
5×7×9	229 (?)	229 (234)	229	2.833717544
5×7×10	254 (?)	254	254	2.835812967
5×7×11	277 (?)	277	277	2.834094219
5×7×12	300 (?)	300 (?)	296	2.826218294
5×7×13	325 (?)	325 (?)	325	2.835070644
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13×14×16	1820 (?)	1820 (?)	1806 (1825)	2.820327226
13×15×15	1803 (?)	1803	1803	2.818128235
13×15×16	1926 (?)	1926 (1932)	1908 (1932)	2.816628414
13×16×16	2038 (?)	2038 (?)	2022	2.815680662
14×14×14	1725 (?)	1725 (?)	1719	2.822787486
14×14×15	1813 (?)	1813	1813	2.818400950
14×14×16	1943 (?)	1943 (?)	1938 (1939)	2.820652346
14×15×15	1905 (?)	1905	1905	2.812696240
14×15×16	2043 (?)	2043 (?)	2016	2.811264261
14×16×16	2170 (?)	2170 (?)	2142	2.811317904
15×15×15	2058 (?)	2058 (?)	2058	2.817336958
15×15×16	2160 (?)	2160 (?)	2155 (2173)	2.812005396
15×16×16	2302 (?)	2302 (?)	2262	2.807630537
16×16×16	2401 (?)	2401 (?)	2304	2.792481250

Coefficient set status

• total schemes: 680 (29 better Strassen)
• ZT schemes: 356 (52.35%)
• Z schemes: 37 (5.44%)
• Q schemes: 287 (42.21%)

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{perminov202558,
    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}
}